MAGNETISM ELECTRON TRANSPORT MAGNETORESISTIVE LANTHANUM CALCIUM MANGANITE

MAGNETISM AND ELECTRON TRANSPORT IN
MAGNETORESISTIVE LANTHANUM CALCIUM
MANGANITE
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF APPLIED PHYSICS
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
G. Jeffrey Snyder
June 1997

© Copyright by G. Jeffrey Snyder 1997
All Rights Reserved
ii

I certify that I have read this dissertation and that in my
opinion it is fully adequate, in scope and in quality, as a
dissertation for the degree of Doctor of Philosophy.
_______________________________________
Theodore H. Geballe
(Principal Adviser)
I certify that I have read this dissertation and that in my
opinion it is fully adequate, in scope and in quality, as a
dissertation for the degree of Doctor of Philosophy.
_______________________________________
Malcolm R. Beasley
I certify that I have read this dissertation and that in my
opinion it is fully adequate, in scope and in quality, as a
dissertation for the degree of Doctor of Philosophy.
_______________________________________
Robert L. White
(Materials Science Department)
Approved for the University Committee on Graduate
Studies:
_______________________________________
iii

Abstract
It is the goal of this thesis to understand the physical properties associated
with the large negative magnetoresistance found in lanthanum calcium
manganite. Such large magnetoresistances have been reported that this
material is being considered for use as a magnetic field sensor. However,
there are many variables such as temperature, magnetic field, chemical
composition and processing that greatly influence the magnitude of the
magnetoresistance. After introducing the problem in Chapter 1, Chapters 2
and 3 describe the materials synthesis and physical property measurements
used in this work. In Chapter 4, the intrinsic magnetic and electron transport
properties of lanthanum calcium manganite are distinguished from those
that depend largely on the chemical synthesis and processing. Chemical
substitution of lanthanum by gadolinium, discussed in Chapter 5, not only
induces ferrimagnetism, but also dramatically alters the electron transport
because of slight structural changes. The physical mechanisms and empirical
relationships found among the resistivity, magnetoresistance and magnetism
in Chapters 3 and 4 are studied in greater depth in Chapters 6 and 7 and
compared with theoretical predictions. This analysis provides a useful
method for predicting the magnetoresistance as a function of temperature,
magnetic field and transition temperature. The related perovskite, strontium
ruthenate, proves to be a model compound for the study of metallic
ferromagnets. The results of this work is presented in two appendices, and
compared with the manganite results throughout the text.
v

Preface
Looking back at the many years at Stanford, there are many people I would
like to thank for helping me along the long, windy path to a Ph.D. thesis. My
dad, papa Schneider, Dr. Demin, and Frank DiSalvo deserve the credit for
getting me interested in science: chemistry, engineering, materials science and
physics.
The graduate first year classes at Stanford would have been far too
unbearable without the support of my first year commiserators Weber, Jim
and Shelly. My time at the Max Planck Institute in Stuttgart, Germany could
not have been more productive or pleasant thanks to Prof. Dr. Arndt Simon,
the whole Abteilung Simon and foreign student ghetto especially Paul Rauch,
Thomas Braun and Chris Ewels.
Having nothing to do with superconductivity, much of my work at
Stanford was outside the KGB headquarters in Ginzton Lab. The materials
synthesis for this project was done at the Center for Materials Research in the
McCullough building. Bob Feigelson and his group deserve a special thanks
for advice and use of equipment, such as the laser heated crystal growth
apparatus. Some of the crystal samples used in this thesis were grown by
Vlad Beffa, a Stanford undergraduate working as a CMR summer student. I
would also like to thank the CMR support staff: Tracy Tingle with the SEM
microprobe, Glen and Waldo for keeping up the x-ray facility, Ann Marshall
for TEM studies, and Thomas Carlson and Mark Gibson for knowing how to
get everything done in McCullough. I would especially like to thank Bob
White, Shan Wang and their students for teaching me about magnetic and
magnetoresistive materials in their group meetings.
Thanks also to the KGB group, especially Ted and Mac for helping me get
started. K. A. Moler performed most of the experiment and much of the
analysis of the heat capacity experiment in Appendix B. Lior Klein was the
vii

viii
inspiration behind the irreversibility line plot in section 3.2.2.2.9. Thanks also
to Khiem, Steve, Daniel & Jenny, Laurent and the rest of the curry club for
great conversations at lunch.
Much of this work was done in collaboration with Hewlett-Packard in Palo
Alto. The MOCVD films of the manganites were made by Ron Hiskes and
Steve DiCarolis. The XAFS studies in Chapter 5 were performed at the
Stanford Linear Accelerator by Corwin Booth and Bud Bridges from U. C.
Santa Cruz.
This work could not have been completed without the support of friends,
housemates, parents but particularly Sossina Ñ thank you very much.
Funding for this research was generously provided by The Fanny and John
Hertz Foundation, the Air Force Office of Scientific Research, and the
Stanford Center for Materials Research under the NSF-MRL program

Table of Contents
Abstract v
Preface vii
Table of Contents i x
List of Figures xiii
List of Tables xvii
1. INTRODUCTION 1
1.1 Motivation 2
1.2 AMnO 3
1.3 Double Exchange 5
2. MATERIALS SYNTHESIS AND CHARACTERIZATION 9
2.1 Sample Preparation 9
2.1.1 Bulk Polycrystalline Samples 9
2.1.2 Single Crystals 10
2.1.2.1 Flux Growth 11
2.1.2.2 Float Zone 12
2.1.2.3 Thin Films 13
2.1.3 Reactive Samples 13
2.2 Characterization 1 4
2.2.1 Elemental Analysis 14
2.2.2 Structural Analysis 14
2.2.2.1 Neutron diffraction 15
2.2.2.2 X-ray diffraction 15
2.2.2.2.1 Powder X-ray diffraction 16
2.2.2.2.2 Single crystal and films 17
2.2.2.3 X-ray Absorption Fine Structure 17
3. ELECTRONIC AND MAGNETIC MEASUREMENTS 1 9
3.1 Transport Properties 1 9
3.1.1.1 Ohm’s Law 19
3.1.1.2 Magnetoresistance 20
3.1.1.3 Drift velocity, mobility, relaxation time and mean free path 21
3.1.1.4 Hall effects 22
3.1.2 Measurement 22
3.1.2.1 Linearity 23
3.1.2.2 Geometry 24
ix
3

x
3.1.2.3 Contacts 26
3.1.2.4 Reproducibility 27
3.1.2.5 Apparatus 28
3.1.3 Analysis 29
3.1.3.1 Metals 30
3.1.3.1.1 Impurity scattering 30
3.1.3.1.2 Electron-electron scattering 31
3.1.3.1.3 Electron-phonon scattering 31
3.1.3.2 insulators/semiconductors 31
3.1.3.2.1 Band insulators/semiconductors 32
3.1.3.2.2 Polarons 33
3.1.3.2.3 Diffusive Conductivity 35
3.1.3.2.4 Variable range Hopping 35
3.1.3.3 Poor Metals / Heavily doped semiconductors 37
3.1.3.4 Phase transitions 38
3.2 Magnetism 3 9
3.2.1 Measurement 39
3.2.1.1 Apparatus 39
3.2.2 Analysis 43
3.2.2.1 Diamagnetism and Paramagnetism 44
3.2.2.1.1 Larmor diamagnetism 44
3.2.2.1.2 Conduction electron diamagnetism 46
3.2.2.1.3 Pauli paramagnetism 46
3.2.2.1.4 Curie paramagnetism 47
3.2.2.2 Ferromagnetism 49
3.2.2.2.1 Weiss molecular field model 50
3.2.2.2.2 Itinerant electron Model 53
3.2.2.2.3 Generalized Model 55
3.2.2.2.4 Critical region 59
3.2.2.2.5 Landau mean field theory 60
3.2.2.2.6 Arrott Plot 61
3.2.2.2.7 The Curie temperature 61
3.2.2.2.8 Spin waves 63
3.2.2.2.9 Irreversibility 66
3.2.2.3 Antiferromagnetism 69
3.2.2.4 Ferrimagnetism 71
3.2.2.4.1 Mean field model for Gd0.67Ca0.33MnO3 72
3.3 Heat Capacity 7 6
3.3.1 Measurement 77
3.3.1.1 Apparatus 78
3.3.2 Analysis 78
3.3.2.1 Electronic specific heat 78
3.3.2.2 Phonon specific heat 79
4. INTRINSIC ELECTRICAL TRANSPORT AND MAGNETIC
PROPERTIES OF LA 0.67 CA 0.33 MNO 3 AND LA 0.67 SR 0.33 MNO 3 MOCVD
THIN FILMS AND BULK MATERIAL 8 0
4.1 Magnetism 8 1
4.1.1 Low Temperature Excitations 83

4.2 Electronic Transport 8 5
4.2.1 Low Temperature Resistivity 86
4.2.1.1 Temperature independent term 89
4.2.1.2 T2 dependent term 89
4.2.1.3 Relationship to magnetism 91
4.2.2 High Temperature resistivity 92
4.2.3 Transport Near TC 93
4.2.4 Hall effect 94
4.2.5 Crystallographic Phase change 96
4.2.6 Small Polarons 96
4.2.7 Colossal Magnetoresistance 97
4.2.8 Domain Boundary Magnetoresistance 99
4.2.9 Low temperature magnetoresistance 99
4.3 Conclusion 9 9
5. LOCAL STRUCTURE, TRANSPORT AND RARE EARTH
MAGNETISM IN THE FERRIMAGNETIC PEROVSKITE
GD 0.67 CA 0.33 MNO 3
xi
101
5.1 Ferrimagnetism 102
5.1.1 Low temperature moment 103
5.1.2 High temperature susceptibility 104
5.1.3 Low temperature susceptibility 105
5.1.4 Near T C magnetism 107
5.1.5 Magnetism model 108
5.1.5.1 Mean Field Model 108
5.1.5.2 Canted antiferromagnetism 109
5.1.5.3 Spin glass magnetism 110
5.1.5.4 Related Compounds 111
5.2 Electronic Transport 112
5.2.1 Magnetoresistance 113
5.2.2 Small Polaron Hopping 113
5.2.3 Variable Range Hopping 114
5.3 X-ray Absorption Fine Structure 114
5.3.1 Relationship of structure to CMR 116
5.4 Conclusion 116
6. MAGNETOCONDUCTIVITY IN LA 0.67 CA 0.33 MNO 3
118
6.1 Anisotropic magnetoresistance 118

xii
6.2 Magnetoresistance models 119
6.2.1 General Model 120
6.2.1.1 Magnetoconductivity model 121
6.2.1.1.1 T > T C regime 123
6.2.1.1.2 T < T C regime 124
6.2.1.1.3 Anisotropic magnetoresistance 125
6.2.1.1.4 Hall Effect 126
6.3 Conclusion 127
7. CRITICAL TRANSPORT AND MAGNETIZATION OF
LA 0.67 CA 0.33 MNO 3
128
7.1 Magnetism near TC 130
7.1.1 Spontaneous magnetization exponent 132
7.1.2 Susceptibility exponent 134
7.1.3 Positive nonlinear susceptibility 135
7.1.4 Additional magnetic interaction 137
7.2 Magnetoresistance 139
7.2.1 Magnetoresistance scaling above TC 142
7.2.2 Magnetoresistance scaling below TC 144
7.2.3 Magnetoresistance scaling at TC 148
7.2.4 Relation to low temperature magnetoresistance 149
7.3 Conclusion 150
Appendix A. Critical Behavior and Anisotropy in Single Crystal SrRuO 3
Appendix B. Magnetic Excitations and Specific Heat in SrRuO 3
References 182
152
173

List of Figures
xiii
Figure 1-1 The Perovskite structure AMnO 3 where A is a mixture of rare earth and
alkaline earth elements e.g. La 0.67Ca 0.33. 3
Figure 1-2 Double exchange and the electronic structure of AMnO 3. 6
Figure 3-1 Effect of changing the scan length for magnetization measurements. A
YIG crystal is used. The three data reduction schemes are also compared. The
results for the full scan algorithm with scan length less than 5 cm are off scale.41
Figure 3-2 Magnetization of YIG sample as it is rotated along the field axis. 4 3
Figure 3-3 Diamagnetic magnetic susceptibility of typical substrates. The increase
in the susceptibility at low temperatures is due to paramagnetic impurities. 4 5
Figure 3-4 Paramagnetic susceptibility and hysteresis loop of a paramagnetic Fe
containing organometallic compound [78]. 4 8
Figure 3-5 Effective paramagnetic moment of Fe in the organometallic compound
[SC(CH 3) 2C(CH 3)NCH 2CH 2CH 2] 2N - FePF - 6 showing a spin transition [78]. 4 9
Figure 3-6 Calculated inverse magnetic susceptibility of SrRuO 3 using the molecular
field model. 50
Figure 3-7 Mean field magnetization calculated in various fields for SrRuO 3 with T C =
165K. Inset show the very small field dependence of the magnetization (forced
magnetization) in this model. 5 2
Figure 3-8 Energy spectrum of magnetic excitations. Spin wave excitations have a
one-to-one dispersion relation while excitations in the Stoner continuum (shaded
region) do not. The intensity of excitations in the Stoner continuum is strongest
where the spin waves meet the continuum. 5 6
Figure 3-9 The inflection T C measured for a SrRuO 3 pellet. For H < 1 Tesla the
inflection T C is within 1 K of the Arrott T C = 163 K. At higher H the inflection
T C increases by only a few degrees. 6 3
Figure 3-10 Correction factor to the T 3/2 contribution of the magnetization in the spin
wave theory due to a magnetic field H . 64
Figure 3-11 Correction factor to the T 3/2 contribution of the heat capacity in the spin
wave theory due to a magnetic field H . 65
Figure 3-12 SrRuO 3 showing spin-glass like irreversibility of zero-field-cooled and
field-cooled measurements in a small field. The field cooled curve may look
saturated, but is actually less than 1/10 saturated at low temperatures. A small
peak is observed in the zero-field-cooled measurement when the reversibility
point is reached. 6 6
Figure 3-13 Initial magnetization of SrRuO 3 pellet at 5 K, after cooling in zero field.
The magnetization follows a “S” shaped curve providing an inflection point. 6 7
Figure 3-14 Magnetic irreversibility line for polycrystalline SrRuO 3. Above the line
the magnetization is reversible, below it is irreversible. The irreversibility
exponent is about 1.5. 6 8
Figure 3-15 Time dependent magnetization of SrRuO 3 pellet at 5 K. The field was
increased from 0 to 100 Gauss. The magnetization follows a Log(time)
dependence. 70
Figure 3-16 Magnetic susceptibility of a Pt containing “CaRuO 3” crystal. The
crystal was aligned with its 2-fold symmetric axis parallel to the applied field has
a susceptibility characteristic of antiferromagnetic moments aligning parallel to
the field, while the 3-fold axis appears to have moments perpendicular to the
field. 71
Figure 3-17 Temperature - tolerance factor phase diagram from reference [100], with
the position of Gd 0.67Ca 0.33MnO 3 indicated. 7 3
Figure 3-18 Calculated Arrott Plot for Gd 0.67Ca 0.33MnO 3 using the mean field model
with T C = 83.3 K. 7 5

xiv
Figure 3-19 High field differential susceptibility for Gd 0.67Ca 0.33MnO 3 calculated using
the mean field model. The maximum is at 11.5 K which is near T Comp = 14.2 K in
this model. 76
Figure 4-1 Magnetization of La 0.67Sr 0.33MnO 3 polycrystalline pellet at 10kOe. Inset a,
magnetization at 100Oe used to determine T C = 375K. Inset b, full hysteresis
loop at 5 K. 81
Figure 4-2 Magnetization of La 0.67Sr 0.33MnO 3 film (LSM1) on LaAlO 3 at 5kOe. Inset,
full hysteresis loop at 5 K of film and (diamagnetic) substrate. 8 2
Figure 4-3 Magnetization of La 0.67Ca 0.33MnO 3 film (LCM15) on LaAlO 3 at 5kOe.
Inset, full hysteresis loop at 5 K of film and (diamagnetic) substrate. 8 3
Figure 4-4 Magnetization of La 0.67(Ca/Sr) 0.33MnO 3 films and polycrystalline samples
showing the T 2 dependence of the magnetization. Inset, same data as a function
of T 3/2 for comparison. 8 4
Figure 4-5 Comparison of the magnetization of La 0.67Sr 0.33MnO 3 with the T 3/2 term
found at low temperatures, and various fits to the magnetization. 8 5
Figure 4-6 Magnetoresistance of La 0.67Sr 0.33MnO 3 polycrystalline pellet and Film
(LSM1). Inset, simultaneous magnetization and resistivity of the film at
20Oersted, along with the magnetoresistance [R(H = 0 kOe)-R(H = 70 kOe)]. 8 6
Figure 4-7 Magnetoresistance of La 0.67Ca 0.33MnO 3 film (LCM17). Inset, simultaneous
magnetization and resistivity at 20Oersted, along with the magnetoresistance
[R(H = 0kOe) - R( H = 70kOe)]. 8 7
Figure 4-8 Low temperature resistivity (in zero field) of La 0.67(Sr/Ca) 0.33MnO 3 films
(LSM1 and LCM10). Solid lines are the fit to R 0 + R 2T 2 + R 4.5T 4.5 up to 250K
and 200K for LSM and LCM respectively. The dashed lines show the constant
and T 2 terms of the best fit. 9 0
Figure 4-9 High temperature resistivity (warming and cooling) of La 0.67Ca 0.33MnO 3 film
(LCM17) and crystal in zero field. Inset a, same data with different abscissa to
compare small polaron and semiconductor models. Inset b, DSC trace of
polycrystalline La 0.67Ca 0.33MnO 3 showing the heat of the high temperature
structural transition. 9 2
Figure 4-10 High temperature resistivity (warming and cooling) of La 0.67Sr 0.33MnO 3
film (LSM1) in zero field. Inset, same data displayed as in Figure 4-9. 9 3
Figure 4-11 Resistance as a function of field La 0.67(Ca/Sr) 0.33MnO 3 films (LSM1 and
LCM19) in the Hall effect configuration at 5 K. The Hall effect is calculated
from the slope of the line indicated (see text). 9 5
Figure 4-12 Colossal magnetoresistive La 0.67Ca 0.33MnO 3 film from [21, 127]. 9 7
Figure 4-13 Simultaneous magnetization and resistivity in a magnetic field of
La 0.67Sr 0.33MnO 3 polycrystalline pellet at 5 K. Data for both increasing and
decreasing field are shown. Inset, Magnetoresistance of La 0.67Ca 0.33MnO 3 film
(LCM10) at 5 K. 9 8
Figure 5-1 Low temperature magnetization of Gd 0.67Ca 0.33MnO 3 measured in a 5 kOe
field and zero field after cooling in a large field (remnant). 103
Figure 5-2 Inverse magnetic susceptibility of bulk Gd 0.67Ca 0.33MnO 3. Solid line is the
high temperature fit to χ = μ eff 2 /(8(T-Θ)) described in the text. 105
Figure 5-3 Low temperature and high-field magnetic susceptibility, χ = (M (60 kOe)-
M (40 kOe))/20kOe, of Gd 0.67Ca 0.33MnO 3 crystal. Inset, hysteresis loop at 5 K. 1 0 6
Figure 5-4 Arrott plot of polycrystalline Gd 0.67Ca 0.33MnO 3 pellet. 107
Figure 5-5 Magnetization and inverse magnetic susceptibility calculated for
Gd 0.67Ca 0.33MnO 3 using the simplified mean field theory described in the text and T C
= 83 K, T Comp = 17 K. The contribution to the magnetization of each sublattice is
shown in dashed lines. 108
Figure 5-6 High temperature resistivity during heating and cooling a Gd 0.67Ca 0.33MnO 3
film, ln(ρ /T ) vs. 1/T . Inset a, comparison with ln(ρ ) vs. 1/T . Inset b,
comparison with ln(ρ) vs. 1/T 1/4 . 111

Figure 5-7 Low temperature resistivity of Gd 0.67Ca 0.33MnO 3 crystal, ln(ρ) vs. 1/T 1/4 .
Inset a, comparison with ln(ρ/T ) vs. 1/T . Solid lines show linear best fit to the
data shown. Inset b, magnetoresistance of a film at 200 K and 300 K; solid line
is the quadratic fit. 112
Figure 5-8 Fourier transform of kχ(k) from (a) Mn K-edge and (b) Gd L III-edge data on
Gd 0.67Ca 0.33MnO 3 . The solid lines are data collected at T = 69 K, while the
triangles (Δ) are data collected at T = 40 K. Agreement between data above and
below T C is well within the errors of the experiment. Transform ranges for the
Gd edge data are from 3.5-12.5 Å -1 and Gaussian broadened by 0.3 Å -1 . Transform
ranges for the Mn edge data are from 3.2-12.5 Å -1 and Gaussian broadened by 0.3
Å -1 . 115
Figure 6-1 High field (longitudinal) magnetoresistance above and below T C for
La 0.67Ca 0.33MnO 3 film. The solid lines show the fit using the indicated equivalent
circuit 123
Figure 6-2 Low field magnetoresistance and magnetization (relative units) of
La 0.67Ca 0.33MnO 3 at 0.9 T C. The sum of the longitudinal and transverse resistances
minimizes the effect of the anisotropic magnetoresistance. 124
Figure 6-3 Hall effect of La 0.67Ca 0.33MnO 3 below (fully magnetized data only) and
above T C. 126
Figure 7-1. M 2 vs. H /M plot for La 0.67Ca 0.33MnO 3 float zone crystal. A mean field
ferromagnet has linear isotherms with a positive slope. The negative slope for T
>T C indicates a faster than linear increase in M (inset) due to a highly unusual
positive non-linear susceptibility χ 3. 130
Figure 7-2. Data from Figure 7-1 (using the same symbols) scaled with β = 0.27 and
γ = 0.90. According to the scaling hypothesis, all the T < T C data should lie on
a single curve while the T > T C data should lie on a separate, single curve. 131
Figure 7-3. Saturation Magnetization, M 0 as a function of temperature for
La 0.67Ca 0.33MnO 3 crystal. At each temperature, the value shown is M extrapolated
to H = 0 as given by the intercept in Figure 7-1.Solid line is fit to M 0(T ) ∝ (1 -
T /T C) β with β = 0.30. 1 3 3
Figure 7-4. Inverse magnetic susceptibility, 1/χ 0 as a function of temperature for
La 0.67Ca 0.33MnO 3 crystal. At each temperature, the value shown is H/M
extrapolated to H = 0 as given by the intercept in Figure 7-1. Solid line is fit to
1/χ 0 ∝ (T /T C - 1) γ with γ = 0.7 and T C = 263K. 136
Figure 7-5. Magnetization in a magnetic field for a La 0.67Ca 0.33MnO 3 polycrystalline
pellet at 0.9 and 1.1 T C. The solid lines indicate the linear regions in each case.139
Figure 7-6. Magnetoresistance of La 0.67Ca 0.33MnO 3 film compared with -M 2 of a pellet,
both at 0.9 T C. The solid line for the magnetoresistance data shows the fit using
the indicated equivalent circuit. The dashed line in the inset compares the
exponential fit. 140
Figure 7-7. Magnetoresistance of La 0.67Ca 0.33MnO 3 film compared with -M 2 of a pellet,
both at 1.1 T C. The solid line for the magnetoresistance data shows the fit using
the indicated equivalent circuit. 141
Figure 7-8. Fitting parameters σ 0 and ρ ∞ for T > T C in a La 0.67Ca 0.33MnO 3 film. The
temperature dependence of these two parameters reflect the insulating behavior of
the material. 142
Figure 7-9. Fitting parameter σ H2 as a function of temperature in a La 0.67Ca 0.33MnO 3
film for T > T C. The temperature dependence of σ H2 and the square of the
susceptibility are the same, indicating a relationship between the
magnetoconductance and M 2 . 143
Figure 7-10. Fitting parameters σ 0 and ρ ∞ for T < T C in a La 0.67Ca 0.33MnO 3 film. ρ ∞ is
governed by the A + BT 2 terms in the resistivity while σ 0 diverges at T C. The
inset shows σ 0 data fit with a (T C - T) 1.8 power law (dashed line), and σ ∝
exp(M /M E) (solid line). The zero field resistivity ρ(H = 0) = ρ ∞ + 1/σ 0 is shown
for comparison. 145
xv

xvi
Figure 7-11. Fitting parameter σ H as a function of temperature in a La 0.67Ca 0.33MnO 3
film for T < T C. The solid line shows the best fit to the data using a critical
exponent of 0.7. 146
Figure 7-12. Magnetoresistance of La 0.67Ca 0.33MnO 3 film at 262 K ≈ T C. The solid line
shows the fit (for the full data on a linear scale) using the indicated equivalent
circuit. 149
Figure A- 1. Magnetization at 5 K of SrRuO 3 single crystal along several
crystallographic directions showing strong cubic but not uniaxial
magnetocrystalline anisotropy. Inset shows the full hysteresis loop of the single
crystal data along with that of a polycrystalline pellet for comparison. 155
Figure A- 2. Arrott Plot of SrRuO 3 single crystal along easy [110] direction. Inset,
critical isotherm (T = 163K ≈ T C) on a log scale fit to M δ ∝ H with δ = 4.2. 1 5 7
Figure A- 3. Zero field magnetization M 0 of SrRuO 3 single crystal along easy [110]
direction. Solid line shows the fit to M 0(T ) ∝ (1 - T /T C) β with β = 0.36. Inset
showing the same data on a log plot. The critical exponent β appears to change
from Heisenberg-like β = 0.39 near T C to Ising-like β = 0.32 as T decreases. 158
Figure A- 4. Zero field inverse susceptibility 1/χ 0 of SrRuO 3 single crystal along
easy [110] direction. Solid line shows the fit to 1/χ 0(T ) ∝ (1 - T /T C) γ with γ =
1.17 and T C = 163.2 K. The inset shows the same data on a log plot. 159
Figure A- 5. Scaled Arrott Plot of SrRuO 3 single crystal along easy [110] direction
with β = 0.36 and γ =1.17. Symbols are the same as those used in Figure A- 2.160
Figure A- 6. Magnetization as a function of temperature of SrRuO 3 single crystal
along easy [110] direction. Inset shows the approximate T 2 dependence of the
magnetization. 161
Figure A- 7. Inverse magnetic susceptibility (1/χ = M /H ) at H = 10 kOe of
polycrystalline SrRuO 3 compared to the single crystal data from Figure A- 4.
The solid line is the straight-line fit with T C = 165K which demonstrates the
slightly positive curvature of the data. 164
Figure A- 8. Variation of the T 3/2 parameter in fitting the magnetization data of single
crystal SrRuO 3 to M = M S (1 - AT 3/2 - BT 2 ) as the fitting range is increased. The
upper inset shows the correlation of the A and B parameters. In the region where
A is relatively stable (around T max = 60 K), A decreases as T max is lowered. The
symbols are the same as those used in Figure A- 9. 166
Figure A- 9. Variation of the T 2 parameter in fitting the magnetization data of single
crystal SrRuO 3 to M = M S (1 - AT 3/2 - BT 2 ) as the fitting range is increased. In
the region where B is relatively stable (around T max = 60 K), B increases as T max is
lowered. 167
Figure A- 10. Variation of the T 2 parameter in fitting the magnetization data of
single crystal SrRuO 3 to M = M S (1 - BT 2 ) as the fitting range is increased. The
parameter B for this fit is more stable and constant than that shown in Figure A-
8. Inset, variation of Θ 2 in a magnetic field. 169
Figure A- 11. Variation of A and B fitting parameters in the hypothetical case where
the true magnetization is given by T 3/2 and T 5/2 terms. 170
Figure B- 1 Heat capacity of SrRuO 3 cooled in zero field (zfc), in an 8 T magnetic
field, and in zero field after being magnetized (rem). Inset, difference between the
heat capacity measured after cooling in zero field with that in 8 T and the
remnant magnetized state. 174
Figure B- 2. Zero field heat capacity data fit with two free parameters, γ and β. Solid
line, including the T 3/2 contribution expected theoretically for spin waves (see
text). Dashed line, without any T 3/2 contribution. 175
Figure B- 3. The linear term of the heat capacity γ as a function of magnetic field.
Each circle is from a single c P,H datum between 4.3 and 5 K with phonons
subtracted: γ(H ) = (c P,H(T) - βT 3 )/T with β = 0.191 mJ/mol·K 4 . γ(H ) for each
square was determined by fitting 15-20 data points between 6 and 12 K to: c P,H(T)

xvii
= γT + βT 3 . The triangles are calculated from the magnetization data of a single
crystal. 176
List of Tables
Table 3-1 Theoretical 3-dimentional critical exponents for different models and
selected experimental values [86, 87]. 6 0
Table 4-1 Physical Properties of Polycrystalline Pellets 8 2
Table 4-2 Magnetoresistance of Annealed Films. 8 9
Table 5-1 Transition Temperatures for Gd 0.67Ca 0.33MnO 3. 107

MAGNETISM AND ELECTRON TRANSPORT IN
1. Introduction
MAGNETORESISTIVE LA 0.67CA 0.33MNO 3
The development of new materials for technological applications has
opened many doors to innovation in the 20 th century. New electronic and
magnetic materials in particular have helped bring about the information
revolution. Much of the progress is due to materials processing.
Technological applications often have strict compositional and
microstructural requirements for their materials. An integrated circuit for
instance must have several compatible semiconductor, dielectric, and
metallic materials with specific properties in precise locations.
Improvements using well understood materials such as these are usually
incremental.
A risky but potentially more revolutionary method for advancing
technologies is to find a different materials which have inherent properties
superior to those currently in use. There are many known materials which
need to be better understood before it would be clear that their use would be a
significant advancement. In some cases a previously unknown class of
compounds (such as the cuprate superconductors) may have to be discovered.
It is also important to consider other aspects of the material, such as chemical
and thermal stability, toxicity and availability.
The study of new materials physics can have different emphasis. Many
physicists are interested in new materials because they can be used to study a
new physical phenomenon. An example of this is the study of heavy
fermion metals and superconductors which have little potential application
1

2 Chapter 1
in themselves, but the physics learned from their study may be quite useful.
Conversely, one can use physics to help understand new materials for
potential applications. The physics may be well established but will give
valuable insight into the uses and limitations of the material. The emphasis
of this dissertation is on the latter: what physics can reveal about a material
as opposed to what the material can tell you about physics.
1.1 Motivation
This dissertation has been motivated by the desire to understand the basic
transport and magnetic properties and the physics behind them in metallic,
ferromagnetic perovskite oxides. The manganites in particular show a wealth
of complex properties. It has been useful to characterize these properties as
either common to ferromagnetic metals in general or unique to the
manganites. For this reason, the study of SrRuO 3 has been quite useful in
understanding the properties of the manganites. For example, SrRuO 3 and
the manganites shows similar magnetic critical behavior and low
temperature magnetic excitations. It has also been advantageous to further
classify the properties of the manganites as those which are intrinsic to the
material and those affected by processing. The physics of the intrinsic
properties are easier to study, while the extrinsic properties can be easily
modified. Once the inherent properties of the material are understood,
properties which depend strongly on processing can then be tuned for used in
a device with particular characteristics.
In this chapter some background on the manganites is presented, focusing
in the recent interest in ÒColossal magnetoresistanceÓ (CMR). Chapter 2
briefly summarizes the materials synthesis and characterization. In chapter 3
the magnetism and transport experimental procedures are given as well as
the pertinent analysis and theory. In chapter 4 the results of the intrinsic
electrical and magnetic properties of La 0.67 Ca 0.33 MnO 3 are presented. In
chapter 5 the ferrimagnetism and structure-transport correlations are shown

A
Introduction 3
Oxygen
Mn
Figure 1-1 The Perovskite structure AMnO 3 where A is a
mixture of rare earth and alkaline earth elements e.g.
La 0.67 Ca 0.33 .
for the related compound Gd 0.67 Ca 0.33 MnO 3 . Chapter 6 introduces the
magnetoconductivity analysis of the magnetotransport phenomena studied
here. In Chapter 7 this analysis is used to examine the relationship bewteen
the magnetization and the magnetoresistance.
1.2 AMnO 3
The R 1-x A x MnO 3 perovskite manganites, where R and A are some rare
earth and alkaline earth elements respectively and 0.2 < x < 0.5, display the
unusual property of being paramagnetic insulators at high temperatures and
ferromagnetic metals at low temperatures [1-4]. Perovskite is the name of the
structure type, Figure 1-1, containing corner sharing MnO 6 octahedra. Both
end members of La 1-x A x MnO 3 are antiferromagnetic insulators [5], but become

4 Chapter 1
ferromagnetic metals upon doping. The theory of double exchange [6-8],
described in section 1.3, has been developed in order to explain this
phenomenon and correctly predicts x = 1/3 to be optimal doping [9]. Recent
calculations show that a second mechanism such as a Jahn-Teller distortion
may be required to explain the magnetoresistance within the double exchange
model [10-12].
Until recently, much of the experimental work on the manganites has
been motivated by their utility as a cathode materials in solid oxide fuel cells
[13]. Thus many compounds of the type R 1-x A x MnO 3+δ have been studied in
polycrystalline form [14-17]. Much has been learned about their defect
chemistry and high temperature electronic and ionic conductivity. Most of
these compounds are not metallic above room temperature but have
electronic conductivity, presumably due to (small) polaron hopping,
sufficient to make good electrodes.
Interest in the perovskite manganites has expanded since their fabrication
as epitaxial thin films [18, 19]. Some films have shown the insulator to
ferromagnetic metal transition at lower temperatures with a large
magnetoresistance near this transition [20, 21]. ΔR/R(H) of greater than 10 6 %
has been reported for fields of several Tesla [22-25]. Since Giant Magneto
Resistance (GMR) films have a ΔR/R(H) of typically 20% (which saturates in a
few thousand Oersted), the manganite films have been proposed as possible
replacements for GMR read heads in the magnetic recording industry.
However, since magnetic recording devices work at room temperature with
low magnetic fields, the temperature range and field sensitivity of the
manganites in their present state do not make them competitive with GMR
materials. Nevertheless, the rather imprecise term "Colossal Magneto
Resistance" (CMR) has been coined for this phenomenon.* However, since it
* It should be noted however, that such a large magnetoresistance is not

Introduction 5
has been widely adopted it will be employed it here where CMR is defined as
ΔR/R(H) > 10. CMR materials often refers to all manganite perovskites.
Although the films are quite stable and the measurements reproducible
even after several months, it is clear that growth and annealing conditions
greatly influence the properties of the manganite films [27]. Furthermore, the
electrical and magnetic properties of the CMR films are often very different
than those of the materials produced by bulk ceramic techniques or single
crystals with the same nominal composition. Thus, in order to understand
these materials, one should distinguish between the properties intrinsic to
perfect crystalline R 1-x A x MnO 3 and those caused by microstructure, strain,
disorder and/or compositional variations.
From the work described in chapter 4, it is concluded that the low
temperature, CMR phenomenon is not intrinsic to the thermodynamically
stable phases with composition La 0.67 Sr 0.33 MnO 3 or La 0.67 Ca 0.33 MnO 3 .
In chapter 5 the effect of the rare earth magnetism is shown for the case
R = Gd in Gd 0.67 Ca 0.33 MnO 3 . The possibility of structural distortions at T C are
considered for this compound.
1.3 Double Exchange
The theory of double exchange is concerned with the exchange process
involving d-band carriers in a mixed valent oxides. First postulated by Zener
3+ 2+
3+ 4+
[6] to explain the properties of ( La1− A )( Mn1− Mn )O
x x x x 3 [1, 2, 4], the theory of
double exchange was formulated by Anderson and Hasegawa [7] and
DeGennes [8]. The compounds at the two ends of the series are
unique to the manganates. Doped EuO and EuS show magnetoresistances
of 10 4 %, using the above definition, and therefore can be considered a
CMR material. Furthermore, it has been shown that in some Chevrel
phase compounds [26], a magnetic field makes the material
superconducting - which would make them Òsuper-magnetoresistanceÓ
(SMR) materials.

6 Chapter 1
e g
t 2g
e g
t 2g
CaMnO 3 T

Introduction 7
For x = 0, the situation is slightly more complex. Most reports claim that
stoichiometric LaMnO 3 is an antiferromagnet insulator [29]. It is apparently
difficult to prepare stoichiometric LaMnO 3 which likes to lose oxygen or be
rich in lanthanum. Off-stoichiometric LaMnO 3 will contain mixed-valent
manganese and could then be metallic and ferromagnetic. Stoichiometric
LaMnO 3 contains entirely Mn 3+ which has 4 d electrons. The first 3 fill the
spin-up t 2g band, as in CaMnO 3 , while the remaining electron half-fills the
spin-up e g band. The e g band is apparently further split, resulting in an
insulator. There are several ways the band could be split, any or all of which
may be the cause of the insulating behavior. First of all, a half-filled band is
susceptible to splitting due to the Mott correlation effect Ð producing a Mott
insulator. Secondly, the structure is not entirely cubic particularly for the end
members. Such a distortion raises the degeneracy of the t 2g and e g orbitals.
This is known as a Jahn-Teller splitting. Finally, the unit cell relevant to the
electronic structure may be doubled, which will split the e g band in half. The
magnetic structure, by virtue of the antiferromagnetism, has a doubled cell,
which may affect the electronic structure.
At finite values of x there will be x holes (or 1 - x electrons) in the spin-up
e g band. These holes should be free to move and provide a large conductivity.
If, however, the intra-atomic exchange, which holds the spins of all the d
electrons on a given ion parallel, is stronger than the Òhopping integral,Ó
then the hopping can only take place between pairs of ions on which the t 2g
spins (Mn 4+ core) are parallel. Otherwise the two sites have different energies.
The difference in energy is proportional to -cos(θ/2), where θ is the angle
between the neighboring core spins. Since free carriers gain kinetic energy by
being itinerant, this provides a type of exchange mechanism which holds the
two core spins parallel. Conversely, the more parallel the core spins are
aligned, the easier it is for the carriers to become itinerant. Since a magnetic
field has a large effect of aligning ferromagnetically coupled magnetic spins

8 Chapter 1
near T C (magnetic susceptibility becomes large), the application of a magnetic
field should increase the conductivity near T C . This gives a simple qualitative
explanation for the large negative magnetoresistance observed near T C .
It has been suggested the double exchange mechanism alone cannot
provide such a large effect on the resistance [10]. It is proposed, that the
electron-phonon coupling which localizes the conduction electrons as
polarons at T > T C , augments the double exchange mechanism to provide the
observed effects [11, 12]. This conclusion is not universally accepted [9, 30-32].
The polaronic mechanism alone may account for similarly large
magnetoresistance in ferromagnetic semiconductors [30, 33, 34]. The stable
state of a electron donor in a ferromagnetic semiconductor can abruptly shift
from being a shallow to a deep donor as the temperature is raised toward T C .
The increasing spin disorder destabilizes the large-radius donor, which
collapses into a well localized small-polaronic donor. The electron-lattice
interaction plays a pivotal role in this phenomenon. The magneto-resistance
arises because the temperature of the donor-state collapse and the
accompanying metal-insulator transition are increased by the application of a
magnetic field. Other explanations for magnetoresistance in ferromagnetic
materials are discussed in chapter 6.
When superexchange is of comparable magnitude to the double exchange,
a canted antiferromagnetic ground state is expected. This is because
superexchange favors an antiferromagnetic ground state with energy
proportional to cos(θ) while the double exchange is proportional to -cos(θ/2).
The minimum of these two energies is in general some θ ≠ 0 [8].

2. Materials Synthesis and Characterization
In order to find or understand new physical phenomena, samples for
measurement need to be made. The advantage of a materials physicist who is
also a materials chemist is that he/she has more control over the material.
There are many synthetic details which may effect the properties. Also, being
able to make oneÕs samples makes it much easier to chose what materials to
study and get the research started rapidly. A materials chemist who is also a
materials physicist understands what properties of the material is of interest
and what measurements are simple enough to characterize the samples so
that chemical improvements can be made efficiently and effectively.
2.1 Sample Preparation
Sample preparation is sometimes viewed as a black art, or as simple as
making breakfast. Indeed some materials require an immense amount of
time and equipment. However, both of these requirements can be limited if
the sample requirements are not too stringent. Much time and effort can be
saved if the type of sample made just exceeds the sample requirements. Bulk
polycrystalline samples are usually quite easy to make while growing crystals
is more risky and time consuming [35]. When looking for isotropic
properties, polycrystalline samples usually suffice.
2. 1. 1 Bulk Polycrystalline Samples
The most widely used method for preparing polycrystalline oxides is the
direct reaction, in the solid state, of a mixture of solid starting materials.
Powder solids are formed which can then be pressed and sintered to form
dense polycrystalline pellets. Even though the desired phase is
thermodynamically favored, solids do not usually react together at ambient
temperature over laboratory time scales and it is necessary to heat the
reactants at high temperatures to overcome the kinetic barriers. For such
9

10 Chapter 2
reactions, the rate limiting step is usually the solid state diffusion of the
cations across the interface between the starting materials. In order to supply
sufficient thermal energy to enable the ions to jump out of their normal
lattice sites and diffuse through the crystal, high temperatures usually greater
than 1000°C are required. Even at these temperatures, diffusion lengths are
usually quite short. To facilitate this process, the starting materials are
usually ground to a fine powder, which both decreases the length the ions
must travel and increases the surface area for reaction. The powders are often
pressed into a pellet before heating to increase the contact between particles.
Reaction times are usually several days and it is best to repeat the process to
insure homogeneous samples.
Starting materials are usually single cation oxides, carbonates, nitrates or
hydroxides Ñ materials which decompose to form oxides when heated.
Carbonates are popular for the alkali and alkaline earth elements because they
are not hydroscopic and therefore can be weighed accurately in air. On the
initial heating or calcination of carbonate containing mixtures, carbon dioxide
is produced and escapes from the solid. This prevents good sintering of the
material into a dense ceramic, requiring an additional heating.
At such high temperatures, the reactivity of the crucible material must be
considered. Common crucible materials for high temperature reactions are
alumina, zirconia, magnesium oxide and platinum. These materials may
contain other impurities to help in their processing. So, contamination of the
desired product by the crucible may not necessarily be by Al, Zr, or Mg.
2. 1. 2 Single Crystals
Many measurements of materials properties are easier to interpret if single
crystal samples are used. Transport measurements, which require a
contiguous transport pathway across the sample, can therefore be greatly
influenced by the presence of grain boundaries, interfacial impurity phases
and voids which obstruct or alter the paths. For instance, an impurity phase

Materials Synthesis and Characterization 11
(of too little volume fraction to detect by many of the characterization
techniques described in section 2.2) located in the grain boundaries may
unknowingly dominate the resistivity.
Furthermore, many properties of interest are tensorial in nature and
therefore have some degree of anisotropy. The properties of polycrystalline
samples are artificially isotropic due to the random orientation of the
crystallites. This may give a easy way to measure the average property of a
material but may be misleading in two ways. First, the physics of the material
may depend largely on the anisotropy. For example, some materials such as
graphite are metallic in one direction and insulating in another. Secondly,
for practical applications most anisotropic materials experience some kind of
orientation during the materials processing e.g. rolling, extruding, thin film
deposition. Thus, the properties of the final product may strongly depend on
the processing, because of the anisotropy.
2. 1. 2. 1 Flux Growth
The use of a homogeneous, amorphous solution may greatly facilitate
formation of the crystalline product, since convection not diffusion will
transport the ions, and the product will form at much lower temperatures
than by solid state reaction. Flux growth can also sometimes yield metastable
phases which are difficult or impossible to prepare by other means. The
solvent or flux can be any material which dissolves and precipitates the
desired material (solute). At the beginning of the crystal growth, the solute is
entirely dissolved in the solvent. The solubility of the solute in the solvent is
then decreased (usually by decreasing the temperature), causing crystals to
nucleate and grow. Knowledge of the solute-solvent phase diagram will help
determine appropriate concentrations and temperature ranges for crystal
growth. The cooling rate and temperature gradient regulate the number and
size of the crystals grown. If a gaseous transporting agent is used (vapor phase
transport), the crystals will not have to be physically separated from the flux.

12 Chapter 2
Otherwise one has to seriously consider how to remove the solvent after the
crystals are grown. One should also be aware of possible contamination in the
crystal by the flux or crucible material.
For instance, crystals of SrRuO 3 and CaRuO 3 can be grown from SrCl 2 or
CaCl 2 molten salts respectively [36]. The magnetic properties of such SrRuO 3
crystals is described in Appendix A. During some of the growths, crystals with
very different morphologies were found. Microprobe analysis showed
significant platinum contamination in these crystals, presumably from the
crucible. The Pt containing crystals from the SrRuO 3 growth were pyramidal
and paramagnetic not ferromagnetic.
The Pt containing crystals from the CaRuO 3 (denoted ÒCaRu/PtO 3 Ó) had
cubooctahedral or rhombohedral morphologies. X-ray diffraction of selected
single CaRu/PtO 3 crystals had a perovskite unit cell. X-ray diffraction of
powdered crystals showed both perovskite and possibly Ca 4 (Ru/Pt)O 6 [37].
This different phase was discovered also from an attempted crystal growth
with CaCl 2 in a Pt crucible.
2. 1. 2. 2 Float Zone
Congruently melting compounds (materials which melt before
decomposing) are ideal for float zone crystal growth. In this method no flux
or crucible is used, preventing possible contamination problems. A
polycrystalline source rod is made stoichiometricly. A molten zone then
slowly moves down the rod. The molten zone is small enough that the
surface tension of the liquid keeps it suspended between the two solid
sections. The composition can be adjusted for growing non-congruently
melting compounds. In this case, the traveling molten zone would have a
composition different (perhaps containing a flux) from the desired material.
The system used for the work reported here, uses a CO 2 laser focused to about
1mm 2 to heat the molten zone.

Materials Synthesis and Characterization 13
2. 1. 2. 3 Thin Films
Related to vapor growth, is the growth of thin films. Thin films are easy
to manipulate and therefore can be much more useful to electronics
technologies than single crystals. The transport of the material to the
substrate usually takes place in the gas phase or a vacuum. Once the atoms or
molecules hit the substrate they diffuse only until they lose energy and are
incorporated into the solid film. These atoms have a relatively short time to
grow crystals and cannot usually return to the vapor phase. This is in
contrast to standard crystal growth which relies on the solid-fluid equilibrium
to grow single crystals. Nevertheless, single crystal films can be grown when
the substrate is itself a single crystal.
The various thin film deposition techniques differ primarily by the way
the material is transported and how it gets into the vapor phase. The most
common methods used in research transport the material in a vacuum or
low pressure gas. The atoms, ions, or small inorganic molecules are ejected
into the gas phase by evaporation, sputtering or laser ablation of a target.
Chemical vapor deposition uses volatile precursors which can easily be
transported in the gas phase to the hot substrate where they decompose to
make the film. Many of the films used in this work were produced using
solid-source MOCVD. In this case, a solid organometallic compound is
evaporated, transported in the vapor phase to the substrate where it
decomposes to form the film. Oxide films can also be produced by first spin-
coating a sol-gel mixture of the desired metal stoichiometry. During heating,
the sol-gel decomposes and crystallizes to form the film.
2. 1. 3 Reactive Samples
Many compounds react with water, oxygen or nitrogen present in air.
Materials such as the subnitrides of barium [38, 39] or alkali intercalated C 60
[40] decompose rapidly in air. Thus the synthesis and analysis of these
materials is much more complicated since it must be done in an inert

14 Chapter 2
atmosphere or vacuum. A glove box or Schlenck apparatus allows the
manipulation of samples in an inert (usually argon gas) atmosphere.
Samples can then be sealed in glass for analysis.
Even if the sample is reasonably stable in air at room temperature, almost
all non-oxides will react with oxygen if heated to high temperatures. Thus
compounds such as sulfides [41, 42] or nitrides [43] must be sealed in glass
before heating to solid-state diffusion temperatures.
2.2 Characterization
2. 2. 1 Elemental Analysis
There are various ways to confirm or measure the elemental composition
of a sample. Most of these methods utilize properties of the core electrons or
nuclei and tend to be insensitive to the light elements. For elements heavier
than neon, there exists several accurate and common procedures: The
electron microprobe detects characteristic X-rays produced after excitation by
an electron beam in an electron microscope. The characteristic light emission
or absorption of gaseous elemental species can be used for elemental analysis
Ð one common method uses an inductively coupled plasma (ICP).
Rutherford back scattering (RBS) detects the weight and depth of atoms by
elastic scattering of alpha particles off their nuclei. X-ray diffraction can in
principle be used for elemental analysis since it gives essentially an electron
density map. Wet chemical methods such as iodometric titration can be used
to measure the oxidation state of individual elemental species, allowing the
inference of the oxygen stoichiometry.
2. 2. 2 Structural Analysis
The physical properties of a material may depend as much on the
structure as the elemental composition: graphite is quite different from
diamond. The regular periodic structure can be determined from the elastic
scattering of neutrons or X-rays. Wavelengths approximately equal to the

Materials Synthesis and Characterization 15
interatomic spacings (measured in ) are needed. Synchrotron and neutron
sources usually have tunable wavelengths as well as high intensity. For
many purposes, however, laboratory scale X-ray diffraction is often adequate.
2. 2. 2. 1 Neutron diffraction
Neutrons, having a magnetic moment, are sensitive to the magnetic
structure as well as the atomic structure. For example, an antiferromagnet
which has a larger magnetic unit cell than the atomic cell, will cause extra
neutron diffraction peaks not seen in X-ray diffraction. Polarized neutrons
are also sensitive to the orientation of these magnetic moments. Since
magnetic measurements only give the net moment, which is often zero for
an antiferromagnet, neutron diffraction is far superior in determining
magnetic structures. Inelastic neutron diffraction can provide further
information concerning the structure dynamics. For instance phonon and
magnon (spin wave) dispersions can be measured.
2.2.2.2 X-ray diffraction
X-ray diffraction (XRD) can be used both to quickly determine which
phases are present in an unknown sample and to perform a detailed
structural investigation. The difference lies more in the sample preparation
and data analysis than the measurement apparatus. Although synchrotron
source x-ray diffraction experiments are quicker, much of what is desired can
be learned from a standard laboratory experiment.
X-rays scatter off the periodic arrays of atoms in a crystal lattice. The
scattered x-rays produce a pattern unique for a particular substance. This
pattern can be used for either phase identification or it can be analyzed to
determine the position of the atoms in the cell. The orientation of the crystal
with respect to the incoming x-rays determines the orientation of the
diffracted x-rays. So, either a single crystal must be precisely oriented to detect

16 Chapter 2
a particular diffraction peak, or many crystallites randomly oriented can be
used to get an orientation independent response.
2.2.2.2.1 Powder X-ray diffraction
In powder x-ray diffraction, the sample consists of many small crystallites
which are assumed to be randomly oriented. This produces rings of diffracted
x-rays, defined by the angle between the incident and diffracted beams, 2θ.
The diffraction condition is determined by the atomic unit cell. In this way,
the structure type of a material and unit cell size can easily be determined. If
more than one type of material is present, the diffraction pattern will be a
superposition of each of the components. If only phase identification is
desired, this often provides enough information. Reference [40] gives a good
example of how this is done.
The intensities of each diffraction line is determined by the atomic
constituents, and their placement in the unit cell. In principle, both of these
can be determined from the intensities. In practice, this is difficult to achieve.
A primary concern is texturing, or preferred orientation of the crystallites. If
the crystallites are plate or needle like, they will tend to lay in the sample
holder in a non-random orientation. This will cause a variation in the
intensities. Also, individual diffraction peaks often overlap one another in a
powder pattern. Since the determination of the structure depends on how
much intensity is associated with each peak, having overlapping peaks
complicates the solution process. It is often easiest to do structural analyses
on single crystal samples.
Powder diffraction sample holders are usually for flat, planar samples. If
the incident angle of the incoming x-rays with respect to the sample plane is
equal that of the diffracted beam being detected, the x-ray beam is partially
focused (Bragg-Botano parafocusing) to give a narrower diffraction peak. For
this reason, the sample plane is usually rotated (by an angle ω = θ) as the

Materials Synthesis and Characterization 17
detector rotates (by angle 2θ). Plate-like crystallites can be highly oriented in
such a sample holder. The Guinier method utilizes a different focusing
technique for samples placed in a capillary tube. This is ideal for air sensitive
samples which can be easily sealed in a tube. Needle like crystallites are easily
oriented when placed in a tube.
2.2.2.2.2 Single crystal and films
Single crystals diffract an x-ray beam to produce spots as opposed to line. A
particular spot appears only when the crystal is in a particular orientation.
For this reason, three additional angles are used to orient the crystal. If the
crystal is very large, it will absorb much of the x-ray intensity. Small crystals,
about 0.2mm diameter, are usually used for structure determination since
they absorb little and have a constant volume of sample in the x-ray beam at
all time. The crystal can be rotated and the intensity of each diffraction spot
measured. This usually provides enough data so that the independent data to
free parameters ratio is about 10. Examples of structure determinations using
single crystal x-ray diffraction can be found in [38, 39, 41, 42, 44-48]. If there are
many smaller crystals in the x-ray beam, they can be effectively ignored by
measuring x-ray intensity only at positions where a diffraction peak from the
larger crystal is expected. This technique is used in [38, 39, 45, 46].
Single crystal films also produce diffraction spots as opposed to lines.
However, the intensities are small compared to those from the substrate. It is
usually only possible (and only of interest) to determine filmÕs unit cell size,
orientation and texturing. The difference in cell size of the film compared to
a bulk sample gives an indication of the strain on the film.
2. 2. 2. 3 X-ray Absorption Fine Structure
X-ray Absorption Fine Structure (XAFS) gives essentially the pair
distribution function of atoms around a particular element in the sample.

18 Chapter 2
Thus XAFS probes the local structure, as compared to X-ray diffraction which
gives average structure over hundreds of angstroms.
X-ray absorption spectra in this study were collected by Corwin Booth and
Bud Bridges from U. C. Santa Cruz in transmission mode on beam line 4-3 at
the Stanford Synchrotron Radiation Laboratory (SSRL) using powder samples
(grain size less than 30μm). The sample temperature was regulated using an
Oxford helium cryostat system within 0.1 K (absolute temperature may be as
much as 2 K warmer). Data for this experiment were collected above and
below T C , at T=69 K and T=40 K. Data reduction and analysis followed
standard procedures reported previously [49].

3. Electronic and Magnetic Measurements
3.1 Transport Properties
Transport properties: Resistivity, dielectric constant, thermalconductivity,
thermopower, Hall effect, etc. are often a prime concern when engineering a
new material for a particular use. Even if the purpose of the material is not
related to transport, the material may still be required to have certain
transport characteristics. For example, the liquid crystal in a liquid crystal
display should have a low electrical conductivity to minimize resistive losses.
3. 1. 1. 1 Ohm’s Law
Most materials, whether metals, semiconductors or insulators, obey
OhmÕs law to a good extent: the current I flowing in a wire is proportional to
the potential drop V along the wire V = IR (linear response). R is the
resistance of the wire and it depends on the size and shape of the wire. One
generally prefers to use intensive quantities to characterize a material. Thus
OhmÕs law can be used to define the resistivity ρ which is defined to be the
proportionality constant between the electric field E and the current density j
that it induces: E = ρj. Since E and j are vectors, ρ is a second order tensor.
Any deviations of OhmÕs law can be easily described by adding terms with
higher powers of j. The conductivity σ is the inverse of the resistivity (σ = ρ -1 )
such that j = σE. In an isotropic or cubic substance the resistivity tensor ρ has
off diagonal elements equal to zero and three equal diagonal elements ρ (a
scalar), then the conductivity tensor σ has the same form with diagonal
elements σ where σ = 1/ρ. For an orthorhombic substance (for which ρ and σ
have no nonzero off diagonal terms in the appropriate coordinate system),
the conductivity along a principle direction is equal to the inverse of the
resistivity along the same principle direction. Because of these simple
19

20 Chapter 3
relationships resistivity and conductivity are commonly described as if they
were scalars. Some other properties of the conductivity and resistivity tensors
are described in section 6.2.1.
3.1.1.2 Magnetoresistance
Magnetoresistance is simply the change in resistivity as the magnetic field
is applied. For nonmagnetic metals, the magnetoresistance ratio ΔR/R is only
a few percent in large ~1Tesla fields. For symmetry reasons discussed in
section 6.2.1, the magnetoresistance is proportional to H 2 for small H in these
metals. Since this magnetoresistance arises from the complicated orbits of the
electrons on the Fermi surface, the magnetoresistance has large
crystallographic anisotropy.
Even in an isotropic conductor, there exists two possible configurations for
magnetoresistance. When the current is parallel to the applied magnetic field
the longitudinal magnetoresistance is measured. If the magnetic field is
perpendicular to the current path, then the transverse magnetoresistance is
being measured.
If the material itself is not isotropic, different crystallographic orientations
will have distinct longitudinal and transverse magnetoresistances. For
example, a cubic material grown as a thin film may have growth induced
crystallographic anisotropy. Since for practical purposes, the current is
usually constrained to the plane of the film, there are three easily reportable
magnetoresistances: longitudinal and transverse magnetoresistance with
both the current and magnetic in the plane of the film, and transverse
magnetoresistance with the magnetic field perpendicular to the film. The
two different magnetic field orientations will also affect the demagnetization
field and domain size and motion which must be taken into account.
Magnetoresistance that is actually proportional to the magnetization
rather than the magnetic field is called anisotropic magnetoresistance (AMR).
In low fields, the magnitude of M does not change, only the orientation. If M

Electronic and Magnetic Measurements 21
is aligned parallel (or antiparallel) to the current, the resistance is different
than if M were perpendicular to the current. This difference is the anisotropic
magnetoresistance. AMR in La 0.67 Ca 0.33 MnO 3 will be discussed further in
sections 6.1.
3. 1. 1. 3 Drift velocity, mobility, relaxation time and mean free path
It is often useful to think of transport in terms of particles moving with an
average velocity in a field because of collisions with the surroundings. Some
of the basic relationships are given below for electrical conductivity, and can
be easily generalized for other transport phenomena. The electrical current j
(charge per cross-sectional area per time) is related to the number density of
carriers n, the charge of each carrier -e (for electrons), and the average drift
velocity v.
j = -nev.
The drift mobility μ (a tensor like the conductivity) is defined by the
relationship between the drift velocity and the electric field E.
v = μE
Thus the conductivity is proportional to the number of carriers and their
mobility
σ = -neμ
The relaxation time can also be used to relate the drift velocity and the
electric field. The electric force accelerates the carrier, of mass m, for an
average time τ between collisions (the relaxation time) to provide the drift
velocity.
v = E
−eτ
τ
so σ =
m
ne
2
m
The mean free path l, the average distance the carrier travels between
collisions, is a useful quantity. If the mean free path is smaller than the
interatomic spacing a, then clearly a model based on atomic collisions is

22 Chapter 3
inadequate. l ≈ a is known as the Ioffe-Regel limit. To calculate l, the average
speed (not velocity) must be known. Unfortunately, this is the Fermi velocity
which depends strongly on the band structure. For an electron gas of density
n with a spherical Fermi surface, the Fermi velocity v F is
v
.
F ≈ 309 3
m n
h
h σ
so that l ≈ 309 . 2 2 .
3 e n
3. 1. 1. 4 Hall effects
The Hall effect is the transverse electric field that is produced by the
presence of a magnetic field perpendicular to the flow of charged carriers.
This can be described by antisymmetric off diagonal terms in the resistivity
tensor which are a function of the magnetic field. The Hall effect is normally
linear with respect to an applied field. For an isotropic substance E y = j x R H H,
where H is the magnetic field, j x is the current along the x direction and E y is
the transverse electric field. R H is known as the Hall coefficient. An internal
magnetic field due to a nonzero magnetization of the material may cause a
Hall effect. This anomalous Hall effect is proportional to the internal
magnetization M, E y = j x R A M. The angle between E and j caused by the Hall
effect is known as the Hall angle.
For simple metals and semiconductors, the Hall effect (combined with
other data) can reveal the sign and density of the charge carriers. The Hall
coefficient for an electron gas is -1/nec. If there is more than one type of
carrier, the Hall effect is the weighted average of the Hall effect of the
individual carriers and becomes nonlinear in H. This nonlinearity can be
used to determine the sign and density of the individual carriers Ð assuming
field independent mobilities and Hall coefficients for each carrier type [50].
3. 1. 2 Measurement
The primary concern for reliable transport measurements of new
materials is the identification and elimination of systematic errors. Thermal
and electrical noise lower the precision but usually not to an important extent

Electronic and Magnetic Measurements 23
when characterizing new materials. The danger is finding a new effect or
result that is reproducible but none the less spurious. Since measurements of
new materials have usually not been previously performed, it is often
impossible to compare with previous results. Some considerations and
solutions are outlined below.
3.1.2.1 Linearity
Most transport measurements assume some sort of linear response.
Conductivity measurements assume I is linearly proportional to V.
Experimentally this is never exactly true. A test for linearity should always be
made before measuring a new sample. The resistance over several orders of
magnitude should be constant within a few percent. Even if I vs. V is linear,
thermal or contact voltages usually add a nonzero offset voltage V 0 so that V
= IR + V 0 . Using wires from the same spool, pure copper contacts and
shielding may help but will not totally eliminate V 0 . This offset voltage is
easily subtracted. The easiest method is by measuring the voltage at I and -I.
The resistance is then R(I) = (V(I) - V(-I))/2. The best method is to take a full I
vs. V curve and measure the slope. This may take more time and therefore
introduce other errors such as those caused by temperature drift. The current
range used for measurement should be well within the Ohmic regime. A
current too small may have V 0 comparable to V; it is best not to rely heavily
on the subtraction of V 0 . Large currents may produce I 2 R heating of the
sample, in which case the measured resistance should look like it has an
additive term proportional to I 2 . AC measurements are ideal for extracting
only the linear response; in this case, the frequency dependence should also
be checked, and compared to the DC value.
Thermopower S measurements similarly assume the linearity of the
voltage with the temperature difference: V = SΔT. These measurements also,
however, have an offset voltage V 0 that should be subtracted.

24 Chapter 3
Hall effect measurements have several assumptions which need to be
considered. First, since it is basically a resistance measurement, the I vs. V
curve should be checked and V 0 subtracted as above. Second, it is often
assumed that the Hall voltage is linear in the applied magnetic field
E y = j x R H H. This is often not the case, as described above (section 3.1.1.4) for
magnetic materials or semiconductors with more than one type of carrier.
Finally, the effect of the magnetoresistance needs to be subtracted.
Traditionally, and theoretically this can be done by balancing the Hall voltage
so that it reads zero at H = 0, and then the magnetoresistance should also be
subtracted for H ≠ 0. In practice, this does not work. There is usually some
magnetoresistive component in the measurement perhaps due to some
anisotropy or inhomogeneity in the material. Since the magnetoresistance is
an even function of H while the Hall effect is an odd function of H, I have
found it easiest to examine the Hall effect by plotting V y (H) - V y (-H) as a
function of H. This give the antisymmetric portion of the Hall voltage
(subtracting the symmetric magnetoresistive contribution) without assuming
a Hall effect which is linear in the applied field H.
3.1.2.2 Geometry
In order to calculate the resistivity from the measured resistance, the
geometric ratio relating the two must be known. Most analyses are based o n
some temperature or field dependence, and therefore it is more important
that this geometric factor does not change during the experiment than it is to
know the value precisely. The geometrical factor can change if the sample
has internal cracks or if the contacts are flowing or cracking.
For resistivity measurements, the resistance of the entire circuit must be
taken into account. A tiny metallic sample may have a resistance much
smaller than that of the contacts or even the wires leading to the voltmeter.
For this reason, four probe measurements are used. A known current is
applied through the sample from two of the contacts. The voltage across a

Electronic and Magnetic Measurements 25
portion of the sample is sensed with two other contacts. The voltmeter has a
high input impedance so that very little current is drawn from the voltage
contacts. If there is almost no current flowing through the voltmeter circuit,
then there is almost no voltage drop across the contacts or contact wires. In
this way, the voltage measured is the voltage across the sample. Since the
voltage contacts are separated from the current contacts, one must be cautious
of the tacit assumption that the current flows uniformly through the entire
sample. If the current avoids the voltage contacts, the results will be
spurious. For high resistance samples, two probe measurements, where the
voltage contacts are the same as the current contacts, can be used. The
resistance due to the contacts and grain boundaries can be determined with
AC impedance spectroscopy. Extremely high resistance measurements can be
done with an electrometer where a constant voltage is applied and the
electrometer measures the tiny trickle of current that passes through.
The simplest geometry is the bar sample. The resistivity of a bar is the
resistance times the cross-sectional area divided by the length between the
voltage contacts. Samples can either be shaped into bars or rods, or films can
be patterned. Patterned films can be ideal for transport measurements since
the geometric ratio is well known, and the effects of the contacts can be
minimized by patterning small contact wires in the film. Unfortunately the
patterning process can damage the films.
For isotropic two-dimensional samples, the physical dimensions (other
than the thickness) need not be measured. In some cases the geometric ratio
can be calculated analytically using conformal mappings. The geometric ratio
has been calculated for a sheet where four collinear and equally spaced
contacts are in the center of a sheet [51]. The Van der Pauw [52, 53]
configuration uses four contacts placed anywhere on the edge. By switching
one of the current and voltage leads, the geometric ratio can be calculated.
The Van der Pauw configuration can also be used for the Hall effect.

26 Chapter 3
3.1.2.3 Contacts
Bad contacts are often the cause of problems in a transport experiment.
Making good contacts is the closest IÕve come to producing fine art in the
laboratory. Contacts need to be strong and durable yet small. The thermal
stresses of temperature cycling can weaken or break a contact. A weak contact
is worse than a broken one because a weak contact always seems to work at
room temperature, give unusual results at some other temperature and then
return to normal at room temperature. A good contact is not delicate, it
should be able to withstand some mechanical stress at room temperature.
Contacts need to be small when it is assumed that they are point contacts
or if the sample is physically small. If the size or shape of the contact affects
the geometry of the measurement (section 3.1.2.2) any sintering, melting,
cracking, etc. of the contact will affect the measurement. Unfortunately, the
smaller the contact, the weaker it usually is.
The contact wires can also cause stress in the contact since they also will
expand and contract with temperature. Contact wires should have bends to
minimize the strain on the contacts. Contact wires should, if possible, also be
plastically deformed so that they will stay at the desired position without any
mechanical force, before the contact material is applied.
Indium metal makes good contacts for measurements at temperatures less
than 400K. Indium is quite ductile and therefore can withstand considerable
strain before breaking. Freshly cut indium can be pressed onto most flat
sample surfaces. The contact wire can then be sandwiched between a second
piece of indium. In this way, contacts a fraction of a millimeter can be made.
For surfaces which are difficult to adhere to, an ultrasonic soldering iron
usually helps. Shiny, ultrasonically vibrating, liquid indium will stick to
almost anything and can be drawn out into long wires. Any yellow, oxidized
surface of the molten indium will hinder the adhesion to a sample.

Electronic and Magnetic Measurements 27
Silver paint contacts can be made quite small if the suspension has the
appropriate viscosity. These contacts can be used to low temperatures but are
often unreliable. I have had better results with the two component silver
epoxy (epo-tek ¨ 417).
Polyimide based silver epoxy (epo-tek ¨ P1011) can be used well above its
recommended maximum temperature of 300°C. It requires only a low
temperature curing and works almost up to the melting point of silver. This
was used for measurements to 1200K. A platinum paste is available for high
temperature contacts that requires a high temperature sintering.
3.1.2.4 Reproducibility
It is obviously important to only report reproducible data. Important data
should always be reproduced and claims about general properties of a
material should be reproduced on other samples or even types of samples. It
is best to simply not collect irreproducible data. If there is anything unusual
in the above checks, it is probably not worth the time collecting the data.
Extensive automation of the data collection system may actually help
increase the consistency of the measurement. If the temperature is changed
in a uniform way, the data tends to be more consistent than if points were
taken at random temperatures. This is probably due to temperature gradients
in the sample or sample chamber, that depend on the heating or cooling rate.
Measurements as a function of temperature should generally be measured
while heating and cooling in case there is some temperature differential
between the sample and the thermometer. Some contacts break easily when
rapidly cooled, requiring the slow cooling of an automated system. With
enough automation so that the experiment can be left unattended for several
hours or days, extra data can be collected. This allows more frequent checks
for linearity and reproducibility.

28 Chapter 3
3.1.2.5 Apparatus
Transport measurements for this work were made in DC allowing a few
seconds for the current and voltages to stabilize. A constant current source
(Keithley model 220 or 224) was used to supply the current and a separate
voltmeter (Keithley model 181, 182, 196 or 197) used to measure the voltages.
Current is not usually measured independently but can be with the addition
of an electrometer into the circuit. A computer controlled the current source,
read the voltages and temperatures and recorded the data. Typically, 5 pairs of
R(I) = (V(I) - V(-I))/2 were measured at slightly different currents (e.g. 20%
variation) to estimate the precision. For resistances greater than 10 6 Ω (often
T < 100K) two point resistance measurements were made using an
electrometer.
Most of the low temperature and all the magnetoresistance and Hall effect
measurements were made in the cryostat of the Quantum Design MPMS 2
SQUID magnetometer described in section 3.2.1.1. The computer that controls
the SQUID magnetometer can also operate subroutines that can control GPIB
compatible devices. These EDC (External Device Control) subroutines can be
quite general, but have a limited number of available commands. It is
necessary to further process the data recorded from the EDC subroutine in
order to use it in a graphing or spreadsheet program. The standard SQUID
magnetometer software controls the temperature and magnetic field. The
thermometer is not very near the sample, so reliable data must be taken after
the temperature has stabilized. This usually takes about 5 minutes for
routine measurements or 30 minutes if any temperature drift is to be
avoided. Since the measurement can run for several days unattended, time is
usually not much of a problem.
A 10 wire transport probe prefabricated at Quantum Design was used for
the measurements. A special tip was made out of 8 bore alumina rod to
provide a surface perpendicular to the magnetic field. Films can then be

Electronic and Magnetic Measurements 29
attached to this surface with double-sided sticky tape to measure the Hall
effect and transverse magnetoresistance. A simple copper plate attached to
the tip of the transport probe provides a surface parallel to the magnetic field
for longitudinal and transverse magnetoresistance measurements. One
limitation of the SQUID magnetometer is its relatively small sample chamber
(8mm diameter).
High temperature (300K < T < 1200K) measurements were performed in a
furnace (in air) with a slowly varying temperature (≈ 1 K/min.). The
temperature was recorded using a Type-K thermocouple positioned a few
millimeters from the sample. The sample holder was made from an 8 bore
alumina rod. The 2 thermocouple wires and 4 platinum wires which contact
the sample were fed through the bores of the alumina rod. The wires are
positioned to facilitate changing the sample and attaching new contacts. The
furnace was programmed to heat and cool using a standard Eurotherm
(model 818) power regulator and temperature controller. Data were
continuously recorded by a computer using LabView software as the furnace
heated and cooled. Lower temperature measurements can be similarly made
by dipping the probe into a dewar of liquid nitrogen or liquid helium.
3.1.3 Analysis
Careful transport measurements can help identify transport mechanisms.
Knowledge of transport mechanisms not only helps us to understand the
microscopic nature of the material but also allows us to predict other
properties using previous theoretical or empirical knowledge of other
materials with the same transport mechanism. For example a few percent
impurity due to processing may not affect the room temperature resistivity of
a metal while greatly decreasing that of a band semiconductor.
Some of the common transport mechanisms and their predicted transport
properties are given below. Since there is not much overlap of mechanisms
or properties between metals and insulators, it is convenient to categorize

30 Chapter 3
materials as metals or insulators-semiconductors. Using the room
temperature resistivity for this definition is attractive since metals are
supposed to have low resistivity while insulators have high resistivity, but
imprecise since a dirty semiconductor may have a lower resistivity than a
poor metal at some high temperature. Even though theoretically metals
have zero or low resistivity as the temperature approaches zero while
insulators have infinite resistivity at T = 0, such a definition is impractical for
materials which undergo a metal-insulator transition. Here, a metal is
defined to be a material with a metallic slope in the resistance vs. temperature
curve: dρ/dT > 0 while a insulator/semiconductor has dρ/dT < 0 This
definition is consistent with most microscopic models of metals and
insulators.
3.1.3.1 Metals
The resistivity of a metal is best described in terms of elastic and inelastic
scattering mechanisms. Without scattering the itinerant electron
wavefunctions would allow electron transport without loss or transfer of
energy. In the relaxation time approximation, MatthiessenÕs rule applies,
which states that the total resistivity is the sum of each of the individual
resistivity contributions. The different scattering mechanisms usually
depend on temperature with a particular power law. Thus the total resistivity
is often a polynomial in T.
3.1.3.1.1 Impurity scattering
Impurities and defects disrupt the disrupt the Bloch wavefunctions and
scatter electrons. Since this perturbation of the crystal lattice is unaffected by
temperature, the resistivity is temperature independent. The impurities can
be in the form of elemental substitutions on an atomic site, interstitial atoms
or vacancies - anything that disrupts the periodic electronic structure. Since a

Electronic and Magnetic Measurements 31
grain boundary is a plane of defects, grain boundary resistance is similarly
temperature independent and often quite large.
3.1.3.1.2 Electron-electron scattering
The strong Coulomb interaction between electrons provides a mechanism
for electron-electron scattering. The exclusion principle limits the scattering
to electrons in partially occupied levels near the Fermi surface. For two
electrons to scatter, both must be in this shell of partially occupied levels with
width k B T about the Fermi level. This leads to a scattering rate and therefore
resistivity proportional to (k B T) 2 .
3.1.3.1.3 Electron-phonon scattering
Other than impurities, lattice vibrations or phonons can disrupt the
periodic structure of the lattice. This will scatter the electrons and lead to an
intrinsic source of resistivity present even in a sample free of crystal
imperfections. The most widely employed expression for this resistivity is
the Bloch-GrŸneisen formula [54]: ρ =
− − − ∫
5
5
KT z dz
6 z z
Θ ( e 1)( 1 e )
D
Θ D
T
0
where K is a
constant characteristic of the metal, and Θ D is the Debye temperature (see
section 3.3.2.2).
At high temperatures T/Θ D > 0.5, this gives a resistivity proportional to T:
2
ρ ≈ KT/4ΘD . At low temperatures T/ΘD < 0.1 the resistivity follows the Bloch
T 5 law: ρ ≈ 124.4 KT 5 6
/ΘD .
3.1.3.2 insulators/semiconductors
A material is insulating or semiconducting when dρ/dT < 0. The
mechanisms described below predict this behavior because the conduction
must be thermally activated. The conducting species must overcome some
energy barrier for conduction. Thermal energy supplies the energy to

32 Chapter 3
overcome the barrier and therefore an increase in temperature increases the
conductivity.
3.1.3.2.1 Band insulators/semiconductors
The electron transport in a semiconductor with an energy gap E g between
the conduction and valence bands is limited by the number of excited carriers.
In the intrinsic regime (low doping concentrations and high temperatures),
the chemical potential lies in the middle of the gap so that the thermal energy
required to excite a carrier is E g /2. The carrier concentration n thus increases
exponentially with temperature n ≈ exp(E g /2k B T). The drift mobility μ = v/E
defined by the drift velocity v in an electric field E in a semiconductor can be
complicated and depends on the doping [55]. At low temperatures, impurity
scattering is expected to dominate giving μ ∝ T 3/2 . Above about 100K lattice
scattering will dominate with an approximate temperature dependence
μ ∝ T -3/2 . Thus the conductivity of an intrinsic semiconductor is σ = neμ ≈
T ν exp(E g /2k B T). For elemental semiconductors, the experimentally observed
mobilities have an exponent ν = -1.5 or larger (more negative) [55]. The Hall
effect is more complicated to determine a priori for an intrinsic
semiconductor since the concentration of electron and hole carriers are the
same.
In the extrinsic regime (doped semiconductors at lower temperatures) the
carrier concentration is determined by the thermal excitation of carriers from
donor or acceptor impurities. The carrier concentration retains the form
n ≈ exp(ΔE/k B T) where ΔE is determined by the difference in energy between
the donor and band energies and the temperature regime of interest [56]. For
example, ΔE is one half the band gap (E g /2) for an intrinsic semiconductor or
ΔE = E g - E d ≈ E g for an extrinsic semiconductor with donor (or acceptor)
energy levels E d from the band edge. Since usually one type of carrier (hole or
electron) is dominant, the Hall effect R H = 1/nec is quite large since n is small,

Electronic and Magnetic Measurements 33
varies exponentially with temperature and can be used to determine the sign
of the carrier and the carrier concentration.
3.1.3.2.2 Polarons
A localized electron will always distort its surroundings relative to an
unoccupied site simply because of the coulombic interaction of the electron
and the surrounding atoms. The potential well produced by this distortion
acts as a trapping center for the self-trapped carrier. The quasiparticle
composed of a self-trapped electronic carrier taken together with the pattern
of atomic displacements that produces the self-trapping became known as a
polaron because self-trapping was first considered in ionic (polar) materials.
The quasiparticle can move as a whole, the electron and the distortion
moving together.
The spatial extent of the self trapped states depends on the range of the
interaction. Long-range electron-lattice interactions [57] produces large
polarons with a finite radius. In such an instance, the self-trapped electronic
carrier extends over multiple sites. The radius of the large polarons decreases
continuously as the strength of the electron-lattice interaction is increased.
The multi-site extension of the large polaron results in its itinerant motion.
In contrast, small polarons are confined to a single lattice position due to
strong short-range electron-lattice interaction. If the electron-lattice
interaction is too weak, the carriers remain unbound. The extreme
confinement of the small polaron typically leads to its moving by thermally
assisted hopping.
One of these two different types of polarons will be stable when both long-
range and short-range electron-lattice interactions coexists. The presence of
long-range interactions eases the requirements for forming a small-polaron,
but only once the interactions are sufficiently strong will the polaron change
from large to small.

34 Chapter 3
The dichotomy between the two types of polarons seems unnecessary
when considering their spatial extent, but the conductivity of the two types of
polarons are vastly different. Large polarons move with significant mobilities
μ > 1 cm 2 /V sec, that decrease with increasing temperature (metal-like). In
contrast, small polarons move with very low mobilities, μ Θ D /2) the electrical
conductivity is predicted [58] to be σ = 3ne 2 ωa 2 /2k B T -1 exp(-E A /k B T). Here n is
the number density of charge carriers, a is the site-to-site hopping distance, ω
is the longitudinal optical phonon frequency and e is the electronic charge.
The Hall mobility may behave like the drift mobility with 1/3 of the
activation energy, but in the adiabatic approximation the form is much more
complicated [58].
In the theory of small polaron hopping, it was found that the motion of
the polaron is only weakly dependent on the magnetic order, and changes
little as the temperature is raised above the Curie temperature [61, 62].
Small polarons were first studied in the non-adiabatic regime, where the
electron transfer integral and hence the bandwidth is small [60, 63-65]. In the
high temperature limit this gives a factor of T 3/2 instead of a T in the drift and
Hall mobilities. There are some unphysical assumptions required for the
non-adiabatic analysis to be valid. The assumptions behind the adiabatic

Electronic and Magnetic Measurements 35
polaron formation are more physically plausible and therefore the adiabatic
analysis is preferred.
3.1.3.2.3 Diffusive Conductivity
The same form of the conductivity derived for adiabatic small polaron
hopping is found more generally for diffusion limited conduction. If the
charge carrier must overcome an activation energy, E A , to hop to a
neighboring site, the probability for hopping will be proportional to
exp(-E A /k B T). From the theory of the random walk, the diffusion constant D
can be estimated using this hopping probability, the frequency, ω, with which
an attempt to hop is made (usually the frequency of the optical phonon which
provides the lowest barrier to hopping at some instant), and the site to site
distance, a: D = λωa 2 exp(-E A /k B T). λ is a geometrical factor approximately
equal to 1. The mobility is related to the diffusivity via the Nernst-Einstein
relation μ = eD/k B T. Thus the conductivity, σ = neμ, of a general, activated,
diffusive process is σ = ne 2 ωa 2 /k B T -1 exp(-E A /k B T). The transport of ions in a
crystal is diffusive, and therefore the ionic conductivity is often analyzed
assuming this form of the conductivity [66]. The transport of small polarons
is very similar and has also been described in this way [67-69].
3.1.3.2.4 Variable range Hopping
For a semiconductor or insulator at low enough temperatures, the
predominant conduction mechanism may no longer be by excitation of
carriers to the mobility edge or by thermally activated hopping to the nearest
neighbor but by variable range hopping. At low temperatures, the
mechanism with the lowest barrier energy will dominate. Due to any
randomness in the sample, the hopping site with the lowest barrier energy
will not in general be the nearest neighbor. The increased hopping distance
will of course reduce the probability that the carrier will tunnel to this

36 Chapter 3
position; however, this is always offset by the lower barrier energy at
sufficiently low temperatures.
The simplest quantitative derivation of the form of variable range
hopping is the following. For a given site, the number of states within a
range R per unit energy is (4π/3)R 3 N(E F ), where N(E F ) is the density of
localized states. Thus the smallest energy difference for a site within a radius
R is on average the reciprocal of this ΔE = 1/[(4π/3)R 3 N(E F )]. Thus, the further
the carrier hops, the lower the activation energy.
The carrier has an electronic wave function exponentially localized on a
particular site with a decay or localization length of ξ. The tunneling
probability that the electron will hop to a site a distance R away will contain a
factor exp(-2R/ξ). The further the distance, the lower the tunneling
probability.
Since the hopping favors large R while the tunneling favors small R,
there will be an optimum hopping distance R for which the hopping
probability proportional to exp(-2R/ξ) exp(ΔE/k B T) is a maximum. This will
occur when 1/R 4 = 8πN(E F )k B T/ξ. Substituting this value for R, the hopping
probability and thus the conductivity is proportional to exp(-(T 0 /T) 1/4 ) where
T 0 = Cξ 3 /k B N(E F ). C is a constant which in this derivation is 24/π ≈ 7.6, but
other values of C are obtained from more sophisticated analyses. C ≈ 21 is
recommended by Shklovskii and Efros.
The conductivity for variable range hopping is usually given the form σ 0
exp(-(T 0 /T) ν ), ν = 1/4. The exact form of σ 0 depends on the model and may
have a power law temperature dependence of its own. For example T 0.33 has
been found [70].
Other values of ν can be obtained theoretically using different
assumptions. The above derivation assumes a three dimensional system. In

Electronic and Magnetic Measurements 37
general, ν = 1/d+1 where d is the dimension of the system. In particular this
gives ν = 1/2 and ν = 1/3 for one and two dimensional systems respectively.
The exponent ν = 1/2 is also found when the Coulomb interaction between
the electrons is taken into account [71]. This interaction produces a gap in the
density of states. If this gap is larger than the bandwidth of the hopping
carriers, ν = 1/2 is found. If polarons are the charge carriers, the range of
validity for ν = 1/d+1 may be reduced to lower temperatures than normally
expected [72].
3. 1. 3. 3 Poor Metals / Heavily doped semiconductors
Some materials which would normally be considered metals, have
dρ/dT < 0 over a significant temperature range. Amorphous metals and
metallic glasses, which can be prepared by rapid cooling, film deposition, or
irradiation, for example, can show a resistivity which decreases slightly with
temperature. Most examples are transition metal alloys with zero
temperature resistivities approaching the Ioffe-Regel limit, where the mean
free path is equal to the interatomic distance, corresponding to about 1 mΩcm
[69]. Heavily doped semiconductors can show similar behavior described
below.
The physics behind this phenomenon is not completely understood, and
the data is usually interpreted on a case by case basis. The theoretical
conductivity is generally described as a power series with respect to
temperature where the power ranges from zero to one, e.g. σ = σ 0 + mT n ,
n < 1, and m can be positive or negative. n = 1/2 or 1 is often observed and
can be explained theoretically. For example, Altshuler and Aronov predict an
n = 1/2 correction for the effect of long-range interaction between electrons
[70]. For semiconductors doped near the metal-insulator transition, n = 1/3
has been observed and explained theoretically [70].

38 Chapter 3
Complex materials often have resistivities that are inadequately described
by the simple models given above. For example the cuprate superconductors
and SrRuO 3 show a linear resistivity well above the Ioffe Regel limit [73]. A
more extreme example is Ba 6 Co 25 S 27 which not only has a resistivity
minimum but a less than linear resistivity above the minimum [42]. As
more materials with complex electronic structures are examined, such non-
standard behavior will certainly become more common.
3. 1. 3. 4 Phase transitions
Any precise measurement as a function of temperature can usually detect
or be influenced by a phase transition. Conductivity data in particular can be
quite precise but can also be greatly influenced by subtle electronic, magnetic
or structural changes. Thus one should be cautious when fitting data to very
similar functional forms. Electronic transitions such as metal-insulator
transitions and charge ordering (charge density wave formation) obviously
change the conductivity at the ordering temperature. Magnetic transitions
are also easily observed in the conductivity since local moments act as
scattering centers, or may even help localize/delocalize carriers.
Ferromagnetic metals for instance always show a decrease in resistivity as the
temperature goes below the Curie temperature [73]. Structural phase
transitions usually change the symmetry and volume, which affects the
conduction paths and density of the charge carriers. This will subtlety change
the conductivity at the transition temperature even if the conduction
mechanism remains the same, as shown in section 4.2.5.
Careful measurements can help determine the nature of such phase
transitions. If the phase transition is reversible and not hysteretic, i.e. the data
are the same upon warming and cooling, then the phase transition is
probably a single second order process. If the data is hysteretic then a first
order process is involved. For example, the ferromagnetic and accompanying
metal insulator transition in the manganites appears to be of second order.

Electronic and Magnetic Measurements 39
The charge ordering transition in the nonmagnetic manganites is also not
hysteretic and probably second order. However, in samples where both
ferromagnetism and charge ordering occur, the conductivity and magnetic
susceptibility are hysteretic, indicating a more complex transition. In this
case, there is no subgroup-supergroup relationship between the two ordered
phases; so to go from the magnetically ordered phase to the charge ordered
phase requires a first order (hysteretic) phase transition [74, 75].
In the region near a second order phase transition (critical regime) physical
quantities such as electronic transport are often best described by a power law
about the critical point, e.g. (T - T C ) n where T C is the critical temperature and n
is the critical exponent. For example, the resistivity of a ferromagnet near the
Curie temperature should be described by a critical exponent (α) equal to that
of the heat capacity [73]. Magnetic critical exponents are described in more
detail in section 3.2.2.2.4
3.2 Magnetism
The magnetism of a material is usually dominated by the electrons near
the Fermi level. Unpaired, localized electrons in insulators have very
different magnetism than paired or itinerant ones. Thus, magnetic
measurements can detect the subtle electronic structure of a material. Since
these measurement do not require contacts, they can be quite easy to perform
Ð and have made them quite standard in characterizing new materials.
3. 2. 1 Measurement
There are various ways to measure a magnetic field and thus
magnetization. The most common instruments to measure D.C.
magnetization are the vibrating sample magnetometer, Faraday balance and
SQUID magnetometer. Surface magnetization can be accurately measured by
the Magneto-optical Kerr effect (MOKE). The apparatus used in this study is a
Quantum Design MPMSR 2 SQUID magnetometer.

40 Chapter 3
3.2.1.1 Apparatus
The Quantum Design SQUID (Superconducting Quantum Interference
Device) MPMS (Magnetic Property Measurement System) is a commercially
available Òturn-keyÓ magnetometer. The device runs on liquid He and can
reach temperatures of 2 K to 400 K and fields up to 7 Tesla. A sample heater is
available to achieve temperatures 300 K to 800 K but the sample space
diameter is reduced from 8 mm to about 3 mm. The revision 2 (MPMSR 2 )
improves the software, allowing simple programming for measurements
lasting several days without requiring maintenance.
The SQUID coils detect the longitudinal component of the magnetization
as the sample is pulled through them. The coils are wound in a second-
derivative configuration in which the upper and lower single turns are
counter wound with respect to the two-turn center coil. This configuration
strongly rejects interference from nearby magnetic sources and allows the
system to function without benefit of a superconducting shield around the
SQUID sensing loops. The raw data are a set of voltage readings taken as a
function of position as the sample is moved upward throughout the sensing
loops. The data are normally fit to a theoretical signal to calculate the
magnetic moment. Several vertical scans are averaged to obtain a standard
deviation.
The two important data reduction algorithms are compared in Figure 3-1
for a standard Yttrium Iron Garnet (YIG) sample. The Òlinear regressionÓ
assumes the sample is properly centered. If the sample is not exactly centered
(longitudinally), the measured moment will be noticeable different. The
Òiterative regressionÓ iterativly finds the center, so if the sample is off-center
the reported moment does not change. When doing temperature scans, the
length of the sample rod will significantly change. The iterative regression is
clearly superior for these measurements. The precision of these two
algorithms are the same (Figure 3-1). The Òfull scanÓ algorithm is simply the

Electronic and Magnetic Measurements 41
sum of the squares of the raw data and does not fit the data to any particular
curve.
Moment (emu)
0.852
0.850
0.848
0.846
0.844
0.842
Scan Length and Algorithm
Iterative
Linear
Full Scan
0.840
2 3 4 5 6 7 8
Scan Length (cm)
Figure 3-1 Effect of changing the scan length for
magnetization measurements. A YIG crystal is used. The
three data reduction schemes are also compared. The results
for the full scan algorithm with scan length less than 5 cm are
off scale.
The distance the sample is pulled through the coils, called the scan length,
can be varied. The effect of the scan length is shown in Figure 3-1. Long scan
lengths are not ideal since field and temperature gradients exist in the sample
chamber. Very long (> 6 cm) will even destabilize the chamber temperature,
requiring several minutes to stabilize before each scan. Field gradients will
cause irreversible flux motion in superconductors - changing the magnetic
moment. Short scans can drastically reduce the accuracy of the moment
without affecting the precision (Figure 3-1). Ferromagnetic and paramagnetic
substances are not very irreversibly affected by field gradients, so a medium
scan length of about 5 cm is used to ensure good accuracy.

42 Chapter 3
The second-derivative configuration of the SQUID coils will, in principle,
not pick up the magnetization of the sample holder if it has a uniform
magnetization. For this reason, a plastic straw with the sample wedged in the
center makes an ideal sample holder. Shrink wrap tubing, shrunk around
the sample also makes a good, but less rigid, sample holder. If string or grease
is used to hold the sample, then there will be an additional contribution to
the magnetization due to the sample holder. Similarly, if the sample is
between two quartz rods, there will be a paramagnetic contribution to the
magnetization due to the gap between the diamagnetic quartz. Materials not
containing transition elements have a diamagnetic susceptibility of about
-0.5 × 10 -7 emu/g, so light sample holders (plastic) are somewhat preferable to
heavier ones (quartz glass).
The measured moment is also somewhat affected by the horizontal
centering of the sample. Rotating the sample along the field axis should have
no effect on the moment. However, it may very slightly change the
horizontal position of the sample. For whatever reason, the moment can
change a few percent with the sample rotator. Such a sensitivity on
orientation probably results in some of the unexpected jumps in the data.
The YIG sample has a moment 96.6% of that expected. Before an
experiment the sample is rotated to give the maximum moment. This is to
minimize the possible underestimation of the moment and usually provides
a region where the moment is less sensitive to small changes of the rotation.
The transverse SQUID coils can in principle also measure the net
transverse magnetic moment. A longitudinal component to the
magnetization will affect the transverse measurement. The sample can be
rotated along the axis of the sample rod, which in combination with the
transverse measurements can be used to measure the transverse-only
moment.

Moment (emu)
0.234
0.233
0.232
0.231
0.230
0.229
0.228
Electronic and Magnetic Measurements 43
Sample Rotation
0.227
0 50 100 150 200 250 300 350
Rotation Angle (°)
Figure 3-2 Magnetization of YIG sample as it is rotated along
the field axis.
3.2.2 Analysis
The important quantities in magnetic measurements are the magnetic
field vector H, measured in Oersted and the magnetization intensity (or
simply the magnetization) M, measured in emu/volume (electromagnetic
unit = erg/Oersted). M is often reported in units of emu/g, emu/mole or
Bohr magnetons (μ B ) per magnetic atom. The magnetic induction field B
measured in gauss, defined by B = H + 4π M, is often considered more
fundamental in physics. The above units are CGS units which are more
prevalent than MKS since the equations tend to be simpler. The MKS unit
which is used commonly in the magnetic literature is the Tesla = 10000 gauss.
Most materials have a magnetization proportional to the applied field.
The magnetic susceptibility χ, defined by M = χH, relates these two quantities.
It is convenient to use χ = M/H, for non-ferromagnetic and isotropic

44 Chapter 3
substances. Otherwise, M = M 0 + χH + χ 3 H 3 + É can be used where χ is now a
second rank tensor describing the linear response, M 0 is the spontaneous
magnetization, and χ n describe the higher order nonlinear terms.
The magnetic field H can be produced either by electric currents or by the
magnetization. If a sample is magnetized, an H field is produced in the
sample to oppose the magnetization and therefore demagnetize the sample.
This demagnetization field H d is for an ellipsoidal sample directly
proportional to the magnetization H d = N d M, where 0 ≤ N d ≤ 4π is the
demagnetization factor. Since the demagnetization field depends on the
shape of the sample, it is the internal field H i = H a - H d the applied field minus
the demagnetization field which affects the magnetic response. Since the
correction for the demagnetization fields is often ignored, one should be wary
of the effects of demagnetization. Ferromagnets in small applied fields for
example may have reduced bulk magnetization due to demagnetization.
3. 2. 2. 1 Diamagnetism and Paramagnetism
Since atoms are made up of charged particles which undergo orbital
motion and have magnetic moments, there are many ways a materials can
have a magnetic response. A diamagnetic response is that for which the
susceptibility is negative while a positive contribution to the susceptibility is
paramagnetic.
3.2.2.1.1 Larmor diamagnetism
Since the electrons in an atom are essentially free charges orbiting a
nucleus, the application of a magnetic field, by LenzÕs law, induces an
opposing magnetic moment. The resulting magnetic susceptibility is
therefore negative and known as Larmor or core-electron diamagnetism.
Both the classical and quantum mechanical analyses yield the same result,
namely that the diamagnetic susceptibility of an atom or ion is proportional
to the number of electrons it contains, Z and its cross sectional area

Susceptibility (10 -6 emu/G g)
0.0
-0.1
-0.2
-0.3
Electronic and Magnetic Measurements 45
χ mol = -ZN A e 2 /6mc 2 where is the mean square electron radius. Since
atomic sizes are roughly the same, and masses nearly proportional to Z, the
diamagnetic susceptibility per gram for all substances are about the same
χ g ≈ -0.5 × 10 -6 emu/g G. Since it is difficult to calculate accurately, it is
best to use experimental values (tabulated in [76, 77]). In most measurements
of paramagnetic and ferromagnetic substances, the diamagnetism is so small
it is usually ignored.
An example of where core diamagnetism is important is shown in
Figure 3-3. Oxide substrates are generally diamagnetic, but provide a
significant susceptibility due to their large mass compared to the mass of a
thin film. The diamagnetism is fairly temperature independent but
substrates may contain paramagnetic impurities which alter the low
temperature diamagnetism.
Diamagnetic Susceptibility of Substrates
LaAlO 3
Al 2 O 3
MgO
-0.4
0 50 100 150 200 250 300 350 400
Temperature (K)
Figure 3-3 Diamagnetic magnetic susceptibility of typical
substrates. The increase in the susceptibility at low
temperatures is due to paramagnetic impurities.

46 Chapter 3
3.2.2.1.2 Conduction electron diamagnetism
The motion of conduction electrons in a metal or semiconductor will also
provide a diamagnetic response (Landau diamagnetism) to a magnetic field.
This is difficult to calculate but generally of the same order as the Pauli
paramagnetism (section 3.2.2.1.3). A superconductor, however, expels a
magnetic field completely (Meissner effect) by the motion of the
superconducting electrons. The diamagnetism of a superconductor is,
therefore, very large, χ vol = -1/4π emu/cc G in the Meissner regime. Since
superconductivity is affected by temperature and a magnetic field, the
magnetic response of a superconductor is actually quite complicated.
3.2.2.1.3 Pauli paramagnetism
Electrons in a metal can be partitioned into spin-up and spin-down bands,
parallel and antiparallel to an applied magnetic field H. The magnetic field
will lower the energy of the spin-up band compared to the spin-down band
(by 2μ B H) and spin-down electrons will flip their spins and pour over into the
spin-up band. The number of electrons (per volume) that need to flip their
spins is approximately the density of electronic states, n(E F ) times one half of
the energy splitting. This produces a net magnetization proportional to the
2
magnetic field and therefore a positive susceptibility, χvol = μB n(EF ). A more
sophisticated statistical-mechanical derivation produces the same result with
small correction proportional to T 2 . The Pauli susceptibility of a metal should
thus be nearly temperature independent and about the same magnitude as
the Larmor diamagnetism. Since the contribution of the Landau
diamagnetism is not known, it is usually not possible to get more than an
order-of-magnitude estimate of the Pauli susceptibility from magnetization
measurements. In principle, however, the Pauli susceptibility should be a
measure of the bare density of electronic states at the Fermi level, n(E F ), in the
absence of many body effects. The linear electronic specific heat term

Electronic and Magnetic Measurements 47
(γ, section 3.3.2.1) is also proportinal to n(E F ), as well as the electronic effective
mass. The ratio of these two experimental results is known as the Wilson
ratio.
3.2.2.1.4 Curie paramagnetism
If a material has unpaired electrons, their magnetic spins will be highly
susceptible to a magnetic field. The energy to orient the spins comes from the
magnetic energy MH which for an atom with effective magnetic moment p, is
μ B pH. The thermal energy k B T, randomizes the spins to oppose the
alignment, so the amount of alignment is roughly proportional to μ B pH/k B T.
The net magnetization is then μ B p times the fraction of moments that are
aligned. So, the magnetic susceptibility of free spins should behave like
2 2
μB p /kBT. This is several hundred times larger at room temperature than the
core or Pauli diamagnetism/paramagnetism, and significantly temperature
dependent.
The full statistical mechanical treatment for quantum mechanical
moments with spin S, orbital angular momentum L, and total angular
momentum J, gives a magnetization per atom M = gμ B JB J (gμ B JH/k B T). g is the
LandŽ g-factor, which for atoms where J = S is the just the electron g-factor,
3 1 ⎡SS
( + 1) − LL ( + 1)
⎤
g = 2.00, otherwise gJLS ( , , ) = +
2 2
⎢
⎣ JJ ( + )
⎥.
1 ⎦
Transition metal ions generally have their orbital angular momentum
quenched so J ≈ S. The Brillouin function, B J (x), is defined by
2J+ 1 x( 2J + 1)
1 x
J + 1
BJ ( x)
= coth − coth . For small x, BJ ( x)≈x , while for
2J
2J
2J 2 J
3J
very large x, BJ (x) ≈ 1. The magnetization is then linear with respect to the
magnetic field until it approaches its saturation value of gμ B J. In the linear
region, μ B H « k B T, the susceptibility has the Curie form χ mol
2 2
NAμ
B gJJ ( + 1)
=
3k
T
B

48 Chapter 3
Susceptibility (emu/G mol)
4.0 10 -2
3.5 10 -2
3.0 10 -2
2.5 10 -2
2.0 10 -2
1.5 10 -2
1.0 10 -2
5.0 10 -3
0.0 10 0
where N
2
AμB 1 emu K
≈
. Since the spin state is not necessarily known, the
3kB
8 Gauss mol
effective moment p is often reported where p 2 = g 2 J(J + 1). The Curie
susceptibility diverges as the temperature approaches zero; however, in real
systems the spins will Òfreeze-outÓ below some temperature making, for
example, an antiferromagnet, ferromagnet, or spin glass.
Magnetization (μ B /Fe)
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-80 -60 -40 -20 0 20 40 60 80
Magnetic Field (kOe)
0 50 100 150 200 250 300 350 400
Temperature (K)
Figure 3-4 Paramagnetic susceptibility and hysteresis loop of a
paramagnetic Fe containing organometallic compound [78].
Insulators with magnetic atoms far apart (do not interact) are good
examples of Curie paramagnets. Figure 3-4 shows the magnetic properties of
an organometallic compound [SC(CH3 ) 2C(CH3 )NCH2CH2NH2 ] -
2 FeCl with well
isolated Fe +3 ions [78]. The magnetization is slightly nonlinear at 5 K, as
predicted by the Brillouin function. The linear susceptibility can be fit to
χ = χ 0 + C/T where χ 0 is the diamagnetic correction (section 3.2.2.1.1). If the

Electronic and Magnetic Measurements 49
diamagnetic correction is assumed to be known, and the effective Curie
moment as a function of temperature can be extracted. The compound
[SC(CH3 ) 2C(CH3 )NCH2CH2CH2 ] 2N - Fe PF -
6 in Figure 3-5 has a gradual spintransition
of the Fe +3 from S = 1/2 to S = 5/2 as the temperature is increased
[78].
Effective paramagnetic moment (μ B )
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
0 50 100 150 200 250 300 350 400
Temperature (K)
Figure 3-5 Effective paramagnetic moment of Fe in the
organometallic compound [SC(CH3 ) 2C(CH3 )NCH2CH2CH2 ] 2N -
showing a spin transition [78].
FePF -
6
3.2.2.2 Ferromagnetism
If the magnetic species interact, then their magnetic properties can be
profoundly altered. Neighboring moments which prefer to be aligned
parallel lead to ferromagnetism while antiparallel alignment gives
antiferromagnetism. More complex interactions lead to ferrimagnetism,
canted ferromagnetism, and spin glass behavior.

50 Chapter 3
H/M (T/μ B )
80
70
60
50
40
30
20
10
5T
5000G
5Gauss
Calculated 1/χ for SrRuO 3
0
160 180 200 220 240 260 280 300
Temperature (K)
Figure 3-6 Calculated inverse magnetic susceptibility of
SrRuO 3 using the molecular field model.
3.2.2.2.1 Weiss molecular field model
The phenomena of ferromagnetism, antiferromagnetism and
ferrimagnetism can often be well described by a mean field or molecular field
model. The molecular field model simply assumes that all the interactions
from the neighboring magnetic species can be described in terms of an
effective internal or molecular field H m , which is proportional to the magnetic
order H m = λM (for convenience in this section, M will refer to the
magnetization per molecule). For a ferromagnet λ > 0. The total magnetic
field on an atom is then the sum of the applied field H a and the molecular
field. Substituting H a + H m = H + λM for H gives a transcendental function of
M in terms of H and T which can be solved iteratively. For a simple
ferromagnet the only parameter is the molecular field constant λ, which can
2 2
be determined from the critical temperature TC via λ = 3kBTC /μB p .

Electronic and Magnetic Measurements 51
Above the Curie temperature the molecular field model gives the Curie-
Weiss law for the zero-field susceptibility χ mol
2 2
NAμ
B gJJ ( + 1)
=
3k
T − Θ , where ΘP is
B P
the paramagnetic Curie temperature. In the Curie-Weiss molecular field
model Θ P = T C . Experimentally, for reasons discussed below, the experimental
value of Θ P is usually a slightly larger than the true critical temperature T C .
In a field, this susceptibility is slightly less than linear (χ 3 < 0), like it is for
paramagnetic moments, due to saturation. The calculated inverse
susceptibility, 1/χ = H/M, is shown in Figure 3-6 for SrRuO 3 . The 1/χ = 0
intercept of the extrapolation of 1/χ vs. T is the paramagnetic Curie
temperature Θ P .
Below T C the mean field magnetization decreases exponentially slowly as
the temperature is raised from T = 0 K, as shown in Figure 3-7. The zero field
magnetization decreases to zero at the ferromagnetic Curie temperature T C .
Also at low temperatures, the mean field magnetic susceptibility is very
small, i.e. the application of a magnetic field increases the calculated
magnetization only slightly (Figure 3-7).

52 Chapter 3
Magnetization (μ B )
Calculated Mean Field Magnetization for SrRuO
3
2.0
1.5
1.0
0.5
Magnetization (μ B )
2.005
2.000
1.995
1.990
0.0
0 50 100
Temperature (K)
150 200
The molecular field model assumes very little about the microscopic
origin of ferromagnetism, only that it makes sense to define a local molecular
field. Microscopic models for the interaction of atomic magnetic moments
can be formulated into a similar mean field model. In this case, the
interatomic exchange energy J can be related to the molecular field constant λ.
A microscopic model also leads to a more accurate prediction of the
magnetization. At low temperatures, the quantized moments can be
collectively excited as discussed in section 3.2.2.2.8 to give the faster decrease
in the magnetization than the molecular field model. The microscopic
model also gives a more realistic description of the observed properties near
the critical point (section 3.2.2.2.4).
Mean Field Forced Magnetization
5K
50K
1.985
0 20 40 60 80 100
Magnetic Field (kOe)
5T
5kOe
5 Oe
Figure 3-7 Mean field magnetization calculated in various
fields for SrRuO 3 with T C = 165K. Inset show the very small
field dependence of the magnetization (forced magnetization)
in this model.

Electronic and Magnetic Measurements 53
3.2.2.2.2 Itinerant electron Model
The ground state properties of a metallic ferromagnet are believed to be
fairly well described by the band theory which has advanced in the recent
decades [79]. The thermal excitations and finite temperature properties,
however, are more difficult because of the effects of strong electron-electron
correlations in narrow energy bands. This limits the utility of traditional
Hartree-Fock or mean field theory described briefly in this section and
requires a treatment of generalized spin density fluctuations described i n
section 3.2.2.2.3 to describe fully the magnetic excitations in itinerant electron
ferromagnets.
The quantum mechanical exchange interaction can lead to
ferromagnetism in an electron gas within the band model, commonly known
as Stoner ferromagnetism [80, 81] or itinerant electron ferromagnetism.
Using the molecular field approximation, a ferromagnetic state is realized for
Jn(E F ) > 1 where J is the average intra-atomic exchange energy per atom and
n(E F ) is the density of states at the Fermi level. This is also known as the
Stoner condition for ferromagnetism. For Jn(E F ) < 1 the material is not
ferromagnetic but has a Pauli susceptibility enhanced by 1/(1-Jn(E F )) [82]. This
factor is called the Stoner enhancement factor.
The effect of the magnetization at finite temperatures can be calculated
using the mean field approximation. Specifically, the energy of the up (down)
spin band is uniformly lowered (raised) by an amount proportional to the
magnetic polarization ζ and a molecular field constant. Since this model
assumes electrons are either in up or down spin bands, the excitations are
known as single particle, spin flip, or Stoner excitations. There is no energy
gap required for Stoner excitations. However, zero or low energy excitations
require a non-zero momentum transfer, while excitations with zero
momentum transfer require an excitation energy equal to the exchange
splitting as can be seen in Figure 3-8. The region of allowed Stoner excitations

54 Chapter 3
is called the Stoner continuum. Collective, spin-wave excitations are not (yet)
included. The mean field analysis [81] gives (M/M(0,0)) 2 = 1-(T/T C ) 2 +
2χ(0,0)H/M.
As expected from a Ginzberg-Landau/mean-field approximation section
3.2.2.2.5, this formula predicts mean field exponents of β = 1/2 and γ = γ ′ = 1
for T near T C , and linear M 2 vs. H/M Arrott plots. Above T C the inverse
susceptibility is proportional to T 2 2
-TC [81], which is in gross disagreement
with the experimentally observed Curie-Weiss law (1/χ = T - T C ). The finite
zero field susceptibility at T = 0, χ(0,0), is essentially the enhanced Pauli-
susceptibility [83].
Unlike the local moment model which requires the zero temperature
saturation moment M(0,0) to be equal to the sum of all the moments, the
band model allows for opposing moments in the minority band at T = 0 (ζ

Electronic and Magnetic Measurements 55
T/T C « ζ 2 (fractional polarization)[81]. A more complete improvement of the
Stoner theory takes into account the effect of spin density fluctuations. Such
improvements are discussed the following section.
3.2.2.2.3 Generalized Model
The self-consistent-renormalization (SCR) theory [79, 84, 85] combines
both single particle and spin wave excitations into generalized interacting
spin density fluctuations. It deals with the coupled modes of spatially
extended spin density fluctuations in a self-consistent fashion. The theory
gives a new mechanism for the Curie-Weiss magnetic susceptibility
associated with the spatially extended modes of spin fluctuations instead of
the local moments as spatially localized spin fluctuations. Furthermore, the
theory is successful quantitatively in explaining various physical properties of
weak itinerant ferro- and antiferromagnets: the theory contains a limited
number of parameters which can be determined experimentally. Thus by the
1970Õs the theory of magnetism was advanced to the stage of having two
established regimes in the opposite extremes, the local moment systems and
weak itinerant ferro- and antiferromagnets. In the former regime the spin
fluctuations are localized in real space while in the latter they are localized in
k-space. In order to unify the two extremes into a single theory an
interpolation theory for the intermediate regime was developed in the 1980Õs
using the functional integral method within the static approximation.

56 Chapter 3
Energy
Magnetic Excitation Spectrum
Spin
Waves
Stoner
Continuum
Momentum Transfer
Figure 3-8 Energy spectrum of magnetic excitations. Spin
wave excitations have a one-to-one dispersion relation while
excitations in the Stoner continuum (shaded region) do not.
The intensity of excitations in the Stoner continuum is
strongest where the spin waves meet the continuum.
The theory predicts that spin waves exist only in a small region at the
origin of (ω,k) space with dispersion relation ω = k 2 D, where D is the spin
wave stiffness. The strongest intensity of the dynamical susceptibility is in
the region where the spin wave excitations meet the Stoner continuum
shown in Figure 3-8. In the low temperature limit a T 3/2 law is predicted i n
the magnetization. However, calculations for Ni 3 Al [86] show that the
magnetization is best approximated by a T 2 dependence over a broad
temperature range below T C . It has been assumed that in the highly
correlated limit, i.e. metallic ferromagnets with a large saturation moment,
the magnetization will obey the spin wave T 3/2 law [85], presumably because
of strong evidence for the existence of spin waves in metallic ferromagnets.

Electronic and Magnetic Measurements 57
The major advancement of the SCR theory is the development of a new
mechanism for the Curie-Weiss magnetic susceptibility applicable to itinerant
electron ferromagnets. The paramagnetic Curie-Weiss susceptibility arises
essentially from the opposition of the magnetic energy to order the spins and
the thermal energy to disorder them. The magnetization is then proportional
to µ B H/k B T, giving the inverse susceptibility H/M linear in T. In the case of
the weak itinerant electron ferromagnets there are no localized spins above
T C so the susceptibility cannot arise from this mechanism. Instead, the
changing mean-square local amplitude of the spin fluctuation provides the
Curie-Weiss susceptibility.
The SCR expression for the inverse magnetic susceptibility contains a
2
contribution from the mean-square local amplitude of the spin fluctuation SL [84] which can be derived from the fluctuation-dissipation theorem.
2
Calculations have shown that SL increases almost linearly with temperature
above T C giving rise to the Curie-Weiss law. This is in contrast with the local
2
moment mechanism where SL is a constant. The Curie constant of the new
mechanism is related to the band structure around the Fermi surface and is
independent of the saturation moment at T = 0. This Curie-Weiss law should
hold even for paramagnetic metals when they are very close to the
ferromagnetic instability, in contrast to that predicted by paramagnon theories
[84].
2
The temperature variation of SL (T), which determines the new Curie-
Weiss mechanism is strongest when the stiffness of the longitudinal spin
fluctuations is small. When the longitudinal stiffness is small, a relatively
2
rapid increase in SL (T) with inverse temperature is expected above TC and it
should saturate at a certain value determined by the band structure. After
saturation the spin fluctuations behave like local moments with a certain
degree of short range order, since the local amplitude of the spin density is
almost fixed. This phenomenon is called temperature-induced local

58 Chapter 3
moments [84]. Thus, a subtle change is expected in the Curie-Weiss
mechanism at some temperature above T C . This has been used to explain the
change in the Curie constant at around 3T C in Co(S/Se) 2 [87]. This
phenomenon is also expected in some exchange-enhanced paramagnets.
In Heusler Alloys such as Pd 2 MnSn, Ni 2 MnSn, Cu 2 MnAl, etc., manganese
atoms are spatially well separated from each other (> 4Å), and are believed to
carry well-defined local moments since the Rhodes-Wohlfarth ratio is close to
one and neutron scattering experiments can be well analyzed with the
Heisenberg model. The ferromagnetic MnPt 3 , FePd 3 and antiferromagnetic
FePt 3 are likely to have well-defined 3d local moments that couple with
itinerant 4d or 5d electrons of Pd or Pt. CoS 2 and the metallic ferromagnets
AMnO 3 described in chapter 4 fit perhaps best in this category.
In the intermediate regime between the local moment ferromagnets and
the weak itinerant-electron ferromagnets are the Ferromagnetic transition
metals (Fe, Co, and Ni) and compounds Fe 3 Pt and CeFe 2 . These materials
have relatively high T C ’s but have Rhodes-Wohlfarth ratios [84, 88] greater
than one. The ferromagnetic oxide SrRuO 3 also fits in this category as
described in detail in Appendix A.
MnSi is a good example of a weak-ferromagnetic helimagnet which is well
described by the SCR theory. Other weak ferromagnets such as Ni 3 Al, ZrZn 2
and Sc 3 In can also be described in terms of the SCR theory [84].
Metallic Cr which orders antiferromagnetically below 312K with a long
period spin density wave, is the most often used example of an itinerant
electron antiferromagnet. However, the Curie-Weiss law is not obeyed above
T N . The magnetism of Cr is best described with an itinerant electron approach
with the addition of a nesting Fermi surface model to describe the spin
density waves [84]. The antiferromagnetic metals γ-Mn, γ-(FeMn) and γ-Fe
have also been studied [84].

Electronic and Magnetic Measurements 59
Examples of nearly ferro- or antiferromagnetic metals which show a large
susceptibility are CoSe 2 (Curie Weiss) Pd or YCo 2 [84], and V 2 O 3 [89]. The
nearly ferro- or antiferromagnetic semiconductor FeSi (Curie Weiss) shows a
Curie-Weiss susceptibility above 700K with a broad transition at 500K to a low
susceptibility state [84].
3.2.2.2.4 Critical region
As the temperature approaches the critical temperature T C , the
spontaneous magnetization vanishes and the susceptibility diverges. This
makes these properteis not analytic near T C . The observed properties are well
described by a power law |T - T C | n where n is a scaling exponent. For
example, the spontaneous magnetization varies as (T C -T) β for T < T C , while
the susceptibility varies as (T C - T) γ for T > T C , and for T = T C (critical isotherm)
M δ = H. The scaling exponents in the critical region are called critical
exponents.
The critical exponents are very similar for a wide range of second order
phase transitions. They tend to depend only on the general form of the
interaction causing the phase transition, such as the dimensionality, and not
on the particular material. Thus most ferromagnets fall into the universality
class of 3-dimentional Heisenberg or Ising ferromagnets. The theoretical
critical exponents calculated for these models is given in Table 3-1. There are
only two unique critical exponents in each universality class. The others are
related via the scaling relations such as γ = β(δ -1).
According to the scaling hypothesis [90], the magnetic equation of state in
the critical region depends only on the scaled variables H/|T C /T - 1| β+γ and
M/|T C /T - 1| β . For example, a plot of the scaled M 2 vs. the scaled H/M, (scaled

60 Chapter 3
Arrott plot), will then have only two curves: one branch for the T < T C data
and another for T > T C .
The magnetization near T C is predicted in the SCR theory, described i n
4/3 4/3 1/2
section 3.2.2.2.3, to behave as M = (TC -T ) [84, 91] which reducecs to β = 1
for T near T C .
Table 3-1 Theoretical 3-dimentional critical exponents for
different models and selected experimental values [92, 93].
β γ δ
Ising .33 1.24 4.8
Heisenberg .36 1.39 4.8
Mean Field .5 1 3
ZrZn 2 .50(3) 1.02(5) 3.1(3)
Fe, Ni, YIG .37(2) 1.2(2) 4(1)
3.2.2.2.5 Landau mean field theory
Near the critical temperature T C the molecular field, or mean field model
(section 3.2.2.2.5) predicts mean field critical exponents (Table 3-1). The
Landau theory of continuous, second order phase transitions (excluding
fluctuations) arrives at the same mean field result. Here the free energy is
expanded in a Taylor series of the order parameter (M in the case of
ferromagnetism). Due to the symmetry of the order parameter, only even
powers of M are nonzero. The first few terms are [94]: G = G 0 + a(T - T C )M 2 +
bM 4 - HM. At a given H and T, M can be found by minimizing the free energy
G. The general solution is H/M = 2a(T - T C ) + 4bM 2 . Below T C the saturation
magnetization (H = 0) is found to be M 2 = (T C - T)a/2b, giving the critical
exponent β = 1/2. Above T C in a field, M is small so the bM 4 term can be
ignored. This gives a susceptibility χ = M/H = (T - T C ) -1 /2a, and a critical

Electronic and Magnetic Measurements 61
exponent γ = 1. Along the critical isotherm T = T C , H = 4bM 3 so the critical
exponent δ = 3.
3.2.2.2.6 Arrott Plot
According to the Landau equation of state H/M = 2a(T - T C ) + 4bM 2 , a plot
of M 2 vs. H/M at a constant temperature gives a straight line. At T C , this line
intersects the origin. The M 2 = 0 intercept gives the inverse susceptibility
1/χ 0 (H = 0) while the H/M intercept gives the spontaneous magnetization
2
M0 (H = 0).
In real systems near the critical point, non mean field critical exponents
are observed. Thus the plot of M 2 vs. H/M (Arrott plot) will not necessarily
be straight lines, but allows a visual analysis of the data [94]. The T C
determined from the Arrott plot is the isotherm that extrapolates to the
origin. Likewise, the extrapolation to the axis intercepts give the H = 0
spontaneous magnetization M 0 (T < T C ) and inverse susceptibility
1/χ 0 (T > T C ) and allows a visual exstimation of the error. From these M 0 (T)
and χ 0 (T) the critical exponents β and γ can be estimated by fitting
M 0 (T) ∝ (1 - T/T C ) β and χ 0 (T) ∝ (T/T C - 1) -γ .
If the assumption of scaling is incorrect, such as a crossover from one
scaling region to another as the temperature changes, then the entire data will
not scale with the same exponents. In this case, the method described above
is better for characterizing the separate regions. Such a crossover is observed
in SrRuO 3 where β appears to change from 0.32 to 0.38 (Appendix A) so that
the entire data does not scale very well with a single average value for β.
3.2.2.2.7 The Curie temperature
Because the Curie temperature affects many of the properties, there are
several ways of measuring it. One of the simplest is finding the paramagnetic

62 Chapter 3
Curie temperature, the temperature where the linear extrapolation of the 1/χ
curve crosses 1/χ = 0. Near the critical temperature however, 1/χ is not
necessarily linear in T since 1/χ = (T - T C ) γ where γ is usually large than 1.
Thus the paramagnetic Curie temperature is usually slightly larger than the
true critical temperature.
In principle the thermodynamic critical temperature can be found by
fitting the divergence of all the measured quantities and using the scaling
relations. This assumes a true critical region has been reached where the
critical exponents do not change. The simpler method using the Arrot plots,
described in the previous section (3.2.2.2.6) can give as accurate results. Thus,
the Arrott T C is probably the best simple way of finding the critical
temperature.
The most basic definition of T C for a ferromagnet is the temperature where
the spontaneous magnetization drops to zero (as the temperature is raised).
Unfortunately due to demagnetization effects, the zero applied field
magnetization will drop to zero at T C faster and less uniformly than expected.
Nevertheless, the temperature where the magnetization, measured in zero or
small applied field, is often used to measure the critical temperature and is
called here the remnant T C .
In an applied field, the magnetization does not drop to zero because of the
large susceptibility near T C . However, below T C the magnetization has a
negative curvature while above T C it has a positive curvature. Thus the
inflection point of the M vs. T curve (denoted here as the inflection T C ) is
often used to measure the Curie temperature. In the mean field model, the
inflection T C is quite close to the real critical temperature. Figure 3-7 shows
the calculated M vs. T of SrRuO 3 in various fields. In a 5 T applied field, the
inflection point is 1.6 K higher than the true T C used for the calculation. This
agrees well with the experimental data shown in Figure 3-9, where the

Electronic and Magnetic Measurements 63
derivatives are calculated numerically from the measured magnetization. In
some systems, the inflection T C can increase substantially as the applied field
is increased, as seen for Gd 0.67 Ca 0.33 MnO 3 in section 5.1.4, making it a poor
estimate of the Curie temperature.
Magnetization (µ B )
1.0
0.8
0.6
0.4
0.2
SrRuO 3 Inflection T C
163 K
167 K
166 K
164 K
5.5T
3T
1T
5kOe
1kOe
300 Oe
50 Oe
0.0
140 150 160 170 180 190
Temperature (K)
Figure 3-9 The inflection T C measured for a SrRuO 3 pellet.
For H < 1 Tesla the inflection T C is within 1 K of the Arrott T C
= 163 K. At higher H the inflection T C increases by only a few
degrees.
3.2.2.2.8 Spin waves
Collective excitations (spin waves) are expected to be the dominant mode
of excitations at the lowest temperatures. This is because it can be shown that
collective excitations will always have lower energy than single spin
excitations. The spin wave theory has been mostly developed for localized
moments [95] but can also be generalized to include k-space electrons [81].

64 Chapter 3
The energy ω of a spin wave is given by ω = Dk 2 + ∆ for low momentum k
spin waves. D is the spin wave stiffness and ∆ is a gap energy arising from
anisotropy or applied magnetic fields. With no gap the spontaneous
magnetization decreases as T 3/2 [95]: M(T)/M 0 = 1 - (T/Θ 3/2 ) 3/2 . For a
3 4
=
2. 612
⎛ nS ⎞ πD
. The
⎝ ⎠ kB specific heat of these excitations [95] also follows a T 3/2 law:
ferromagnet with spin S and spin density n, Θ3/ 2
c V = 1.925 k B nS(T/Θ 3/2 ) 3/2 . The spin wave stiffness D may be estimated from
the critical temperature T C and number of near neighbors z, using D = 2JS 5/3
and k B T C /J = 0.0521(z - 1)(11S(S + 1)-1) [96], where J is the exchange energy.
Correction factor M(t)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0.0 0.5 1.0 1.5 2.0
0.0
0 2 4 6 8 10
k B T/gµ B H
Figure 3-10 Correction factor to the T 3/2 contribution of the
magnetization in the spin wave theory due to a magnetic field
H.
The first order effect of a magnetic field providing a gap ∆ = gµ B H /k B T, is
to adjust the T 3/2 coefficient by a function of k B T/gµ B H: M(T)/M 0 =
1 - F 3/2 (k B T/gµ B H)/ζ(3/2) (T/Θ 3/2 ) 3/2 . This correction factor M(t), shown in
2

Electronic and Magnetic Measurements 65
Figure 3-10, depends on the temperature T, thus altering the temperature
dependence of the magnetization to have an effective power higher than 3/2
[97]. The Bose-Einstein integral function [95] F P (x) is given by F P (x) =
and ζ(P) =
−
e
n
∞ n x
∑ P
n=
/
1
∞
1
∑ (e.g. ζ(3/2) = 2.612 and ζ(1/2) = -1.460). The correction C(t) to
1 n P
n=
the T 3/2 contribution to the heat capacity is similar: C(t) = [F 5/2 (t) + 4F 3/2 (t)/5t +
4F 1/2 (t)/15t 2 )]/ζ(5/2), where t = k B T/gµ B H. The correction factor C(t) is shown
in Figure 3-11. Due to the Maxwell relation the increase in the effective
power in M(T) also increases the effective power of dc/dH (T).
Correction Factor C(t)
1.0
0.8
0.6
0.4
0.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0 0.5 1.0 1.5 2.0
0.0
0 2 4 6 8 10
k B T/gµ B H
Figure 3-11 Correction factor to the T 3/2 contribution of the
heat capacity in the spin wave theory due to a magnetic field
H.

66 Chapter 3
3.2.2.2.9 Irreversibility
Below T C , a ferromagnet must undergo a first order phase transition when
the field direction is changed. Like all first order processes, this involves
nucleation and growth (of magnetic domains) and can be hysteretic.
Technically there is only hysteresis when the internal field changes sign; if
the field increases or decreases while retaining the same sign (direction), the
change in the magnetization is continuous. However, in real systems, there
exist internal anisotropy and demagnetization fields. So in low fields
irreversible behavior can be found.
Magnetization (µ B )
0.12
0.10
0.08
0.06
0.04
0.02
SrRuO 3 Pellet irreversibility
Field Cooled
20 Oersted Magnetization
Zero Field Cooled
FC
ZFC
0.00
0 50 100
Temperature (K)
150 200
Figure 3-12 SrRuO 3 showing spin-glass like irreversibility of
zero-field-cooled and field-cooled measurements in a small
field. The field cooled curve may look saturated, but is
actually less than 1/10 saturated at low temperatures. A small
peak is observed in the zero-field-cooled measurement when
the reversibility point is reached.
The field required to reduce to zero the magnetization of a saturated
sample, is called the coercive field. When the coercive field is large, such as

Electronic and Magnetic Measurements 67
in polycrystalline SrRuO 3 , the irreversibility will be evident in many
measurements. The coercivity depends on the microstructure of the sample,
and therefore depends on the processing. SrRuO 3 single crystal for instance
(Appendix A) has a coercive field 100 times smaller than polycrystalline
SrRuO 3 .
Magnetization (µ B )
0.8
0.6
0.4
0.2
0.0
SrRuO 3 initial magnetization
Inflection Point
0 1000 2000 3000 4000 5000
Magnetic Field (Gauss)
Figure 3-13 Initial magnetization of SrRuO 3 pellet at 5 K, after
cooling in zero field. The magnetization follows a “S” shaped
curve providing an inflection point.
In the irreversible regime, a ferromagnet will have magnetic properties
similar to that of a spin glass [98]. For example, it is common to characterize
spin glasses [99] (or disordered superconductors) by comparing the field-cooled
(sample cooled below T C in the measuring field) and zero-field-cooled
(sample cooled in zero field and then the field applied before measuring upon
warming) magnetization in a small field. For a spin-glass or disordered
superconductor, these two curves are different. In such small fields, a

68 Chapter 3
ferromagnet will also show such hysteresis, as demonstrated in Figure 3-12
for SrRuO 3 . The field cooled curve may look like a ferromagnetic M vs. T
curve, however it is influenced greatly by the demagnetization field and
coercivity. For example in applied fields lower than the maximum
demagnetization field (4πNM S ) the net magnetization can only increase until
the demagnetization field cancels the applied field (M = H/4πN). The a plot
of M vs. T in this case will show a constant, low value for M, until the true
magnetization decreases near T C (see for example the low field data in [29]).
Reversible Field (H rev ) (Gauss)
100000
10000
1000
100
SrRuO 3 Irreversibility Line
Hrev (const T)
Hrev (Const H)
H rev = (1-T/T C ) 1.5
10
0.01 0.1
1-T/Tc
1
Figure 3-14 Magnetic irreversibility line for polycrystalline
SrRuO 3 . Above the line the magnetization is reversible,
below it is irreversible. The irreversibility exponent is about
1.5.
Another characteristic of a spin glass is an “S” shaped magnetization
curve, which gives an inflection point in M vs. H. Such a curve is also
observed for a ferromagnet, such as that seen for SrRuO 3 in Figure 3-13. The
curves for increasing and decreasing fields will meet at the irreversibility
point. The irreversibility points measured by the two methods (constant T,

Electronic and Magnetic Measurements 69
Figure 3-13; and constant H, Figure 3-12) give an irreversibility line, much
like that in superconductors. For polycrystalline SrRuO 3 the reversible field is
approximately proportional to (1 - T/T C ) 1.5 .
Like many glassy systems, the magnetic properties in the irreversible
region are time dependent. In a ferromagnet the domain walls are pinned
and so must overcome some barrier energy to move. Thus the
magnetization increases in a time dependent manner. Figure 3-15 shows the
magnetization of a SrRuO 3 pellet at 5 K after increasing the magnetic field
from 0 Gauss (where the magnetization was 0.0076 µ B ) to 100 Gauss. The
magnetization increases with a Log(time) dependence, with 98% of the
increase happening before the first data point (about 1 minute). Such a
logarithmic time dependence is also characteristic of a spin glass [99].
SrRuO 3 has all the properties of a ferromagnet, and because of this long
range order can not be a spin glass. The domain structure of a ferromagnet
provides magnetic disorder on a larger length scale than a spin glass, but to be
a spin glass, the disorder must be on the atomic scale.
3.2.2.3 Antiferromagnetism
In an antiferromagnetic substance, the neighboring magnetic moments
align antiparallel. Such a material is called an antiferromagnet. In the
ordered state, there is often no net moment in zero field. Thus magnetization
measurements cannot easily measure the magnetic order. In neutron
diffraction, the magnetic unit cell multiplies giving extra diffraction peaks.
The intensity of these peaks is a direct measure of the ordering.
In a magnetic field, an antiferromagnet has a small positive susceptibility.
Above the ordering temperature, or Néel temperature T N , the susceptibility
can be well described by the Curie-Weiss law used for ferromagnets. Since the
interactions are antiparallel, the coupling is of the opposite sign as that for a

70 Chapter 3
Magnetization (µ B )
0.01645
0.01640
0.01635
0.01630
0.01625
0.01620
Magnetization Creep
100 1000
Time (s)
10000
Figure 3-15 Time dependent magnetization of SrRuO 3 pellet
at 5 K. The field was increased from 0 to 100 Gauss. The
magnetization follows a Log(time) dependence.
ferromagnet. This leads to a negative paramagnetic Curie temperature
Θ P ≈ -T N , where the susceptibility above T N is given by χ mol
2 2
NAµ
B gJJ ( + 1)
=
3k
T − Θ .
B P
Below the ordering temperature T N , M vs. H is still linear with no H = 0
magnetization, but the susceptibility depends on the orientation of the
magnetic field with respect to that of the magnetic moments [56]. For
example, Figure 3-16 shows the susceptibility of platinum containing
“CaRuO 3 ” crystals, or CaRu/PtO 3 described in section 2.1.2.1. If the magnetic
field is applied parallel to the ordered moments, the applied torque (µ × H) is
small so the susceptibility decreases to zero. The CaRu/PtO 3 crystal when
oriented with the 2-fold symmetric axis parallel to H shows such a
susceptibility. When the field is perpendicular to the ordered moments, the
torque is maximized providing a large, constant susceptibility in the ordered

χ (emu/g/G)
4.0 10 -5
3.5 10 -5
3.0 10 -5
2.5 10 -5
2.0 10 -5
1.5 10 -5
1.0 10 -5
Electronic and Magnetic Measurements 71
5.0 10 -6
0 50 100 150 200 250 300 350 400
Temperature (K)
state. This is seen in the crystals oriented with a 3-fold symmetric axis parallel
to H. Without Pt, CaRuO 3 has an antiferromagnetic like susceptibility
(1/(T + Θ P )) above 50K (Θ P ≈ -210K p eff ≈ 9.75 or S ≈ 1.14) but does not become
antiferromagnetic. Instead, it remains paramagnetic but with a stronger
temperature dependence ≈ 1/T (Figure 3-16). From 2 K to 20 K the
susceptibility fits well to (1/(T + Θ P )) with an antiferromagnetic slope
(Θ P = -73K p eff = 4.17 or S = 0.64).
CaRuO 3 and Pt Containing CaRuO 3 Crystal
CaRuO 3
3-fold axis crystal
2-fold axis crystal
Figure 3-16 Magnetic susceptibility of a Pt containing
“CaRuO 3 ” crystal. The crystal was aligned with its 2-fold
symmetric axis parallel to the applied field has a susceptibility
characteristic of antiferromagnetic moments aligning parallel
to the field, while the 3-fold axis appears to have moments
perpendicular to the field.
3.2.2.4 Ferrimagnetism
If the sublattices of different moment orientation do not exactly cancel,
there is then a small ferromagnetic moment in the ordered state, that behaves

72 Chapter 3
with temperature and field much like a ferromagnet. When this is due to a
canting of neighboring moments from truly parallel or antiparallel, this is
called a canted antiferromagnet. If the ferromagnetic moment is not due
entirely to canting but arises from the different strengths of the sublattice
magnetizations, the material is called a ferrimagnet.
Most commercial insulating “ferromagnets” are actually ferrimagnets.
Common examples are ferrites and garnets such as YIG. The multiple
magnetic interactions inherent to a ferrimagnet provide a variety a magnetic
behavior. For example, when the magnetizations of the opposing sublattices
exactly cancel, the net moment is zero like it is in an antiferromagnet. The
temperature where this occurs is called the compensation temperature T Comp .
3.2.2.4.1 Mean field model for Gd 0.67Ca 0.33MnO 3
The gross features of a ferrimagnet can be explained using the molecular
field model of section 3.2.2.2.5. For a ferrimagnet with two sublattices, the
total magnetization M is the sum of the two sublattice magnetizations M A and
M B . The molecular field acting on each sublattice can then arise from
interactions with atoms in the other sublattice as well as atoms in the same
sublattice. For example on the A sublattice H mA = λ AA M A + λ AB M B . The
molecular field constants λ ij can be either positive for ferromagnetic coupling
between sublattices or negative for antiferromagnetic coupling. For a
ferrimagnet or antiferromagnet, there must be strong antiferromagnetic
coupling. Similarly, on the B sublattice H mB = λ BB M B + λ AB M A . Once the
molecular field constants are known, the magnetization of each sublattice can
be calculated iteratively as described in section 3.2.2.2.5 until self consistency is
reached.

Gd 0.7Ca 0.3MnO 3
Electronic and Magnetic Measurements 73
Hwang et al.
Figure 3-17 Temperature - tolerance factor phase diagram
from reference [100], with the position of Gd 0.67 Ca 0.33 MnO 3
indicated.
This has been done for Gd 0.67 Ca 0.33 MnO 3 where the experimental results
are given in chapter 5. In Gd 0.67 Ca 0.33 MnO 3 the Mn atoms are on one
sublattice (B) and Gd atoms on the other (A). The dominant molecular field
constant is expected to be the ferromagnetic Mn-Mn coupling due to the
stronger 3d exchange interaction. From the trend found in the manganese
ferromagnetic ordering temperature as a function of the average of the R and
A atom size [100] in R 1-x A x MnO 3 , shown in Figure 3-17, the T C is expected to be
around 60K. The molecular field constant can then be approximated using
the relationship found for a ferromagnet in section 3.2.2.2.5, namely
2 2
λBB = 3kBTC /µ B
pB .
2
Here pi also includes the site occupancy for each type of atom on the i
sublattice. For Gd0.67Ca0.33MnO3 , the Mn (B) sublattice has two magnetic
contributions since the site is 67% occupied by Mn +3 (S = 2) and 33% occupied

74 Chapter 3
by Mn +4 2
(S = 3/2). This gives pB = Σ nig 2 si (si +1) = 21. On the Gd/Ca (A)
sublattice, only Gd +3 2
(67% occupied, S = 7/2) has a moment giving pA = 42.
The Gd-Gd interaction is expected to be negligible, and therefore λ AA ≈ 0,
since the Gd 4f electrons are well localized. The Gd atoms are expected to
order due to the significant Gd-Mn interaction which is antiferromagnetic, so
λ BA < 0. In the case of Gd 0.67 Ca 0.33 MnO 3 , there exists a temperature where the
magnetization of the larger, but poorly ordered, Gd moments exactly cancels
that of the Mn sublattice. This temperature is called the compensation
temperature T Comp .
The compensation temperature T Comp can be used to estimate the
molecular field constant λ BA . The weakly coupled Gd approximately act like
free paramagnetic spins reacting to the internal (molecular) field caused by
the ordered Mn moments. In zero applied field this gives
2 2
MA = HmA µ B
pA /3kBT. The internal field on the A sublattice is related to λBA from H mA = λ BA M B . At T = T Comp , there is no net magnetization so M B + M A = 0.
2 2
Combining these relations gives λBA = -3kBTComp /µ B
pA .
Above TN , the susceptibility can be calculated analytically [101] giving a
hyperbola for 1/χ vs. T shown in Figure 5-5. The high temperature asymptote
is a straight, Curie-Weiss like, line which intersects the T axis at
2 4 2 2 2 2
ΘP = (µ B /3kB ) (pB λBB + 2pB pA λBA )/(pB + pA ). For Gd0.67Ca0.33MnO3 , λBA < 0 and
2
λBB > |λBA | so ΘP should be less than but close to 7λBB (µ B /3kB ) which is 1/3 of
the value that would be expected (TC of Mn) if no Gd moments were present.
In this model, the ferrimagnetic Néel temperature is given by
2 2
TN = pB λBB (µ B /3kB )[1/2 + √(1/4 + (pAλBA /pBλBB ) 2 )] [101]. For λBB > |λBA | this is
2 2
only slightly greater than the TC = pB λBB (µ B /3kB ) expected if the Gd moments
were absent. Combining this result with that of the last paragraph gives a

Electronic and Magnetic Measurements 75
2 2 2
general requirement of this model TN /ΘP > (pB + pA )/pB = 3, which should be
easily verified from experiments (section 5.1.5.1).
Magnetization 2 (µ B /mole) 2
1.2
1
0.8
0.6
0.4
0.2
Calculated Gd 0.67 Ca 0.33 MnO 3 Arrott Plot
80
83K
85K
0
0 2000 4000 6000 8000 10000 12000
H/M (Oe/µ B )
Figure 3-18 Calculated Arrott Plot for Gd 0.67 Ca 0.33 MnO 3 using
the mean field model with T C = 83.3 K.
The critical temperature T C , of a ferrimagnet is the Néel temperature T N .
As T approaches T N from below, the zero field magnetization vanishes much
like that of a ferromagnet. The theoretical Arrott plot T C (shown in Figure 3-
18) is equivalent to the remnant T C = T N . The mean field calculation gives a
nearly linear critical isotherm on the Arrott plot, shown in Figure 3-18, since
M δ = H and δ = 3 is the mean field critical exponent. The inflection point in
the M vs.. T curve is also a good estimate of T C . In this model, the inflection
T C increases by only 0.55 K/Tesla as H is increased.
Below T C the calculated magnetic susceptibility is small since the internal
molecular field is quite large. The ferrimagnet susceptibility calculated here,
however, is not uniformly decreasing as T decreases like it does for a

76 Chapter 3
ferromagnet. Instead, it peaks as shown in Figure 3-19 near the compensation
temperature T Comp .
Susceptibility (µ B /T)
0.5
0.4
0.3
0.2
0.1
Calculated Susceptibility of Gd 0.67 Ca 0.33 MnO 3
χ = (M(6T) - M(4T))/2T
0
0 20 40 60 80 100
Temperature (K)
Figure 3-19 High field differential susceptibility for
Gd 0.67 Ca 0.33 MnO 3 calculated using the mean field model. The
maximum is at 11.5 K which is near T Comp = 14.2 K in this
model.
3.3 Heat Capacity
The heat capacity of a sample, particularly at low temperatures, is a
powerful experimental technique which gives fundamental information
about low-lying excitations of many kinds (phonons, electrons, magnons,
etc.). Microscopic models which describe these excitations typically give
quantitative predictions of the specific heat which can then be used to check
the theory. The has been quite successful at low temperatures where the
excitations are simple. At higher temperatures, the dispersion relation of the
excitation spectrum can become less simple and therefore complicate the heat

Electronic and Magnetic Measurements 77
capacity. Thermal measurements at higher temperatures T>100K are typically
only used to study phase transitions.
3. 3. 1 Measurement
The most straightforward method of measuring heat capacity is adiabatic
calorimetry. Here, the heat capacity C=∆Q/∆T is measured directly by applying
a known ∆Q and measuring ∆T. This technique requires that the thermal
relaxation time constant between the sample and its surroundings be large
compared to the measurement time. In a common Differential Scanning
Calorimeter (DSC) or Differential Thermal Analyzer (DTA) the loss of heat
due to thermal relaxation is accounted for by subtracting the signal of an
empty sample holder measured at the same rate. A DSC actually operates by
recording the heat input required ∆Q /dt to keep the temperature changing at
a constant rate ∆T/dt = constant. The DTA supplies a constant ∆Q and
measures ∆T between the sample and a reference. These commercial
instruments are used mostly to detect and estimate the entropy associated
with a phase transition. Very accurate measurements require a more
complicated apparatus with better thermal isolation.
The relaxation technique and ac methods for measuring heat capacity are
preferred when measuring a small ∆Q or ∆T. The heat ∆Q is actually
calculated in the relaxation technique by measuring the thermal time
constant of the system. The sample is attached to an object with high thermal
mass and known temperature (the surroundings) by means of a weak
thermal link. After heating the sample, the temperature of the sample
exponentially decays to that of the surroundings. The heat capacity is then
the product of the time constant of this decay and the thermal conductance of
the weak link. This thermal conductivity can be measured in the steady state
by applying a fixed power and measuring the resulting temperature of the
sample.

78 Chapter 3
3.3.1.1 Apparatus
The apparatus, method of operation, data reduction and analysis used at
Stanford is described in detail in [102] and references therein. The sample is
attached to a bolometer by means of Wakefield thermal compound. The
bolometer is a patterned sapphire substrate containing two phosphorus doped
silicon thermometers (used for different temperature ranges) and a heater
with aluminum contact pads (6). Gold-7%Cu wire of 0.001” diameter is
spotwelded to these contacts to mechanically, electrically, and thermally
connect the sample containing bolometer to the surrounding sample holder.
The gold wire provides the weak thermal link to the surroundings and has a
thermal conductivity approximately linear in temperature. The sample
chamber is evacuated before the measurement and care must be taken to keep
out He since it adds to the thermal conductivity and even the heat capacity if
it condenses on the bolometer. The heat capacity due to the sapphire
bolometer, thermal compound, gold wire and aluminum contacts (in order of
decreasing importance) must be subtracted from the raw heat capacity data.
The precision of the measurement is typically 0.5% with or without an
applied magnetic field.
3.3.2 Analysis
The specific heat at low temperatures is usually of the form c = γT + βT 3 .
Because of this, the data are usually plotted as c/T vs. T 2 giving a straight line
with slope β and intercept γ.
The specific heat of spin waves are discussed in section 3.2.2.2.8.
3. 3. 2. 1 Electronic specific heat
The linear term comes from the specific heat of itinerant electrons and is
proportional to the density of electronic states at the Fermi level, n(E F ).
2 2
γ = n(EF )kB π /3. Insulators and semiconductors have no free electrons and
therefore no linear term. Good metals tend to have broad bands with low

Electronic and Magnetic Measurements 79
density of states, giving a γ of the order 1 mJ/mol/K 2 . Bad metals tend to
have narrow bands and high density of states providing the region of large γ
between semiconductors and metals.
3. 3. 2. 2 Phonon specific heat
The cubic term in the specific heat arises from the excitation of phonons in
the crystal lattice, and can be related to the Debye temperature Θ D :
-3
β = ΘD nkB 12π 4 /5. The Debye temperature varies like the spring constant
between neighboring atoms in the crystal: with Θ D being very large for light,
strongly bonded atoms such as diamond, and small for heavy, weakly bonded
atoms such as lead metal. Graphite, which has strong bonding in only two
dimensions, has a specific heat over a significant temperature range which is
proportional to T 2 instead of T 3 .
The Debye model does not account for the specific heat of optical phonons.
At higher temperatures this may become important. Using the Einstein
model the additional specific heat is proportional to nk
Θ E is the Einstein temperature.
B
2 Θ E ( ΘE
/ T) e
Θ E / T
( e − 1)
/ T
2
where

4. Intrinsic Electrical Transport and Magnetic Properties of
La0.67Ca0.33MnO3 and La0.67Sr0.33MnO3 MOCVD Thin Films and
Bulk Material
In this study, measurements among polycrystalline pellets, single crystals
and thin films, supplemented with literature data when available, are
compared to establish the properties inherent to La 0.67 AE 0.33 MnO 3 where AE =
Ca, Sr, Ba or Pb. Whereas most reported La 0.67 AE 0.33 MnO 3 films have been
grown by laser ablation, it is shown here that organometallic chemical vapor
deposition (MOCVD) can produce high quality films. It will be demonstrated
that singular behavior can be obtained in bulk and thin films. Most of the
work described in this section has been previously published [103].
To form the thermodynamically stable phase, the material is annealed at
high temperatures for long times and cooled slowly. This produces films,
polycrystalline pellets and single crystals with very similar properties.
Properties which are common to all these samples are called “intrinsic” to the
thermodynamically stable phase. The remaining differences can then be
attributed to inhomogeneities, noncrystallinity, microstructure, strain or
growth induced effects. Major changes in the magnetization and electrical
transport have been observed in RE 0.67 AE 0.33 MnO 3+δ compounds (where RE is
a rare earth element) when the oxygen stoichiometry was varied; these have
been discussed mainly in terms of non-stoichiometry doping and carrier
localization [14-17, 104, 105].
In order to compare experimental properties certain terms need to be
defined. For simplicity, T C the used in this chapter is the temperature where
the bulk of the zero field magnetization disappears. T MI (metal to insulator
temperature) is defined here as the temperature where the resistivity is a
maximum. T MR is the temperature where the magnetoresistance (R(0)-R(H))
80

Magnetization (emu/g)
80
60
40
20
Intrinsic Properties of La 0.67 Ca 0.33 MnO 3
Magnetization (emu/g)
100
50
0
-50
La 0.67 Sr 0.33 MnO 3 Polycrystalline Pellet
0
0 50 100 150 200 250 300 350 400
Temperature (K)
is the largest. The term “CMR” is used here when the magnetoresistance
ratio ∆R/R(H) is greater than 10 (or 1000%).
b
5K
-100
-75 -50 -25 0 25 50 75
Magnetic Field (kOersted)
10 kOersted
4.1 Magnetism
The polycrystalline samples have relatively square hysteresis loops (Figure
4-1), with forced magnetization at 70kOe of only a few percent. The saturation
magnetizations are close to that expected for high spin manganese in
octahedral coordination: for spin only (orbital contribution quenched)
moment µ = gsµ b , g=2 and then µ= 2µ b [0.67 x 2 (from Mn 3+ ) + 0.33 x 3/2 (from
Mn 4+ )] = 3.67µ b . The measured ferromagnetic and Curie temperatures are the
same within experimental error of a few degrees. The physical properties of
the polycrystalline materials summarized in Table 4-1 Physical Properties of
Polycrystalline Pellets are consistent with previous results [1-4, 29, 106, 107].
10
8
6
4
2
a
100 Oe
0
370 372 374 376378 380
Temperature (K)
Figure 4-1 Magnetization of La 0.67 Sr 0.33 MnO 3 polycrystalline
pellet at 10kOe. Inset a, magnetization at 100Oe used to
determine T C = 375K. Inset b, full hysteresis loop at 5 K.
81

82 Chapter 4
M/M 0
1.0
0.8
0.6
0.4
0.2
Since the precise oxygen concentration has a noticeable effect on T C and M S
[104], slight differences are expected. Float zone grown crystals have similar
magnetic properties.
M/M 0
1.0
0.5
0.0
-0.5
La 0.67 Sr 0.33 MnO 3 Film
-1.0
-75 -50 -25 0 25 50 75
5 kOersted
0.0
0 50 100 150 200 250 300 350 400
Temperature (K)
Table 4-1 Physical Properties of Polycrystalline Pellets
TC MS a
La0.67Sr0.33MnO3 376K
3.59µb /Mn
3.88Å
La0.67Ca0.33MnO3 270K
3.39µ b /Mn
3.86Å
HC 20Oersted 10Oersted
Curie Weiss µ eff
Hall carrier density (of films)
5.61µ b
2.1holes/Mn
5.96µ b
0.9holes/Mn
5K
Magnetic Field (kOersted)
Figure 4-2 Magnetization of La 0.67 Sr 0.33 MnO 3 film (LSM1) on
LaAlO 3 at 5kOe. Inset, full hysteresis loop at 5 K of film and
(diamagnetic) substrate.

Intrinsic Properties of La 0.67 Ca 0.33 MnO 3
The transition temperatures of the annealed films are close to those of the
polycrystalline material, and are summarized in Table 4-2. The saturation
magnetization, measured at 5kOe, also decreases as T 2 for both samples. The
annealed films exhibit a (5 K) coercivity of about 100 Oe and a sheared
hysteresis loop expected from demagnetization or uniaxial anisotropy effects
(Figure 4-2 and Figure 4-3).
M/M 0
1.0
0.8
0.6
0.4
0.2
0.0
M/M 0
1.0
0.5
0.0
-0.5
La 0.67 Ca 0.33 MnO 3 Film
5K
-1.0
-75 -50 -25 0 25 50 75
Magnetic Field (kOersted)
5 kOersted
0 50 100 150 200 250 300 350 400
Temperature (K)
Figure 4-3 Magnetization of La 0.67 Ca 0.33 MnO 3 film (LCM15) on
LaAlO 3 at 5kOe. Inset, full hysteresis loop at 5 K of film and
(diamagnetic) substrate.
4. 1. 1 Low Temperature Excitations
The saturation magnetization decreases approximately proportional to T 2
at low temperatures (Figure 4-4). This temperature dependence of M can be
expected for metallic ferromagnets as discussed in section 3.2.2.2.2.
83

84 Chapter 4
M/M 0
1.00
0.96
0.92
0.88
0.84
1.0
0.9
La 0.67 (Ca,Sr) 0.33 MnO 3 Magnetization
0.0 0.1 0.2 0.3 0.4
(T/T ) 3/2
LCM Pellet (10 kOersted)
LCM Film (5 kOersted)
LSM Pellet (10 kOersted)
LSM Film (5 kOersted)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
(T/T )
C 2
Figure 4-4 Magnetization of La 0.67 (Ca/Sr) 0.33 MnO 3 films and
polycrystalline samples showing the T 2 dependence of the
magnetization. Inset, same data as a function of T 3/2 for
comparison.
A T 2 dependence of the magnetization is predicted by the Moriya theory of
spin fluctuation in itinerant electron ferromagnets, as shown by a calculation
for the weak metallic ferromagnet Ni 3 Al [86]. A T 2 dependence has been
observed in bulk SrRuO 3 (Appendices A and B) which is a metallic
ferromagnet with a substantial saturation moment (i.e. p sat /p cw > 0.5) like the
manganites. In the simplest model, a T 2 dependence is due to single (k-space)
particle spin-flip excitations, while a T 3/2 law is expected for collective
excitations (spin waves). At low temperatures, the T 3/2 law may be seen in
La 0.67 Sr 0.33 MnO 3 [97] with a stiffness (D = 154 meV or Θ 3/2 = 2224 K) also
observed in neutron work. This low temperature form is compared in Figure
4-5 with the corresponding data of Figure 4-4 to demonstrate the limitations
of this form at higher tem-peratures. The next order term in the spin wave

Intrinsic Properties of La 0.67 Ca 0.33 MnO 3
theory predicts a T 5/2 con-tribution. Fitting the T 5/2 parameter Θ 5/2 , along with
the T 3/2 term fits the data for La 0.67 Sr 0.33 MnO 3 better up to 300K than a simple
T 2 fit. Some samples, such as that shown in Figure 4-5, show an increase in
the magnetization at 50 K.
Magnetization (emu/g/G)
90
85
80
75
70
65
La 0.67 Sr 0.33 MnO 3
M/M = 1 - (T/Θ )
0 2 2
Θ = 540K (free parameter)
2
M/M 0 = 1 - (T/Θ 3/2 ) 3/2 - (T/Θ 5/2 ) 5/2
Θ 5/2 = 471K (free parameter)
D = 154 meV
Θ 3/2 = 2224 K
60
0 50 100 150 200 250 300
Temperature (K)
Figure 4-5 Comparison of the magnetization of
La 0.67 Sr 0.33 MnO 3 with the T 3/2 term found at low temperatures,
and various fits to the magnetization.
4.2 Electronic Transport
Both La 0.67 Sr 0.33 MnO 3 and La 0.67 Ca 0.33 MnO 3 have an abrupt drop i n
resistivity below the magnetic Curie temperature in both films and bulk
samples (Figure 4-6 and Figure 4-7). The magnetoresistance (Defined as R(0)-
R(H)) peaks near the Curie temperature as shown in Figure 4-6 and Figure 4-7
and previous experiments ([29, 104]).
85

86 Chapter 4
Resistivity (10 -3 Ω cm)
6
5
4
3
2
1
Resistivity (10 -3 Ω cm)
5
4
3
2
La 0.67 Sr 0.33 MnO 3 Magnetoresistance
La 0.67 Sr 0.33 MnO 3 Film
R (H=0)
∆R/R
M
340 345 350 355 360 365 370 375 380
Temperature(K)
Pellet 0kOe
Pellet 70kOe
Film 0kOe
Film 70kOe
0
0 50 100 150 200 250 300 350 400
Temperature (K)
Figure 4-6 Magnetoresistance of La 0.67 Sr 0.33 MnO 3
polycrystalline pellet and Film (LSM1). Inset, simultaneous
magnetization and resistivity of the film at 20Oersted, along
with the magnetoresistance [R(H = 0 kOe)-R(H = 70 kOe)].
4. 2. 1 Low Temperature Resistivity
Low temperature resistivity data were analyzed using a polynomial
expansion in temperature T. Since the resistivity is essentially constant for
temperatures less than 10K for all samples, it is clear that the resistivity data
both with and without an applied field require a temperature independent
(R 0 ) term. As shown previously for related compounds [29, 104, 108], R 0 is a
function of magnetization in polycrystalline La 0.67 AE 0.33 MnO 3 (Figure 4-8) as
discussed in section 4.2.8. In contrast, the magnetoresistance of films and
crystals at 5 K is over 100 times less: it is not even measurable in the
La 0.67 Ca 0.33 MnO 3 crystal where the precision is 0.5%. Due to the higher
resistance of the films, the magnetoresistance at 5 K is measurable and is
28
26
24
22
Magnetoresistance (%)

Resistivity (10 -3 Ω cm)
10
8
6
4
2
Resistivity (10 -3 Ω cm)
10
9
8
7
6
5
Intrinsic Properties of La 0.67 Ca 0.33 MnO 3
La 0.67 Ca 0.33 MnO 3 Magnetoresistance
La 0.67 Ca 0.33 MnO 3 Film
Magnetization
0 kOe
∆R (0-70 kOe)
240 245 250 255 260 265 270
Temperature(K)
0 kOe
70 kOe
0
0 50 100 150 200 250 300 350 400
Temperature(K)
Figure 4-7 Magnetoresistance of La0.67Ca0.33MnO3 film
(LCM17). Inset, simultaneous magnetization and resistivity at
20Oersted, along with the magnetoresistance [R(H = 0kOe) -
R( H = 70kOe)].
approximately linear with field (Figure 4-8). This effect is small enough for
the analysis to be carried out with R 0 independent of magnetic field.
Since polycrystalline material contains a significant contribution to the
resistivity from grain or domain boundaries, as shown by microwave
measurements [109-111] and higher R 0 values, analysis of the resistivity data
is made only for the well annealed films and crystals. Data up to 70K and
100K for La 0.67 Ca 0.33 MnO 3 and La 0.67 Sr 0.33 MnO 3 , respectively, are well described
with an additional (R 2 ) term proportional to T 2 (Figure 4-8). A linear term in
T did not significantly improve the fit and was not used in later calculations
in order to keep the number of free parameters to a minimum. The field
Magnetization (relative units)
87

88 Chapter 4
dependence of R 2 is so small (less than 10% variation) it has not been
measured.
The range of validity can be extended to about 200K for La0.67Ca0.33MnO3 and 350K for La0.67Sr0.33MnO3 with the addition of a T n term where 4 < n < 5.
The data was analyzed with a fit to R 0 + R 2 T 2 + R 4.5 (H)T 4.5 since a T 4.5
temperature dependence has been predicted for electron-magnon scattering in
the double exchange theory [112]. Evidence for a T 4.5 term (as opposed to T 4 or
T 5 ) cannot be claimed from this data, only that it allows the determination of
the R 0 and R 2 terms to higher accuracy in a much larger temperature range.
The maximum temperature for which the data were analyzed was
determined by finding the maximum temperature where the free parameters
remained stable and the fit was visibly accurate.
The results of this analysis are given in Table 2. As expected, R 4.5 (H)
decreases as the magnetic field increases. In a field of 70 kOe, R 4.5 (H) decreases
by 25-50% (Table 2).
Recently other authors have analyzed the resistivity data for
RE 0.67 AE 0.33 MnO 3 compounds using models different than the one presented
above. The resistivity of polycrystalline La 0.67 Ca 0.33 MnO 3 has been fit to R 0 +
R c T c [29] while a polynomial fit such as the one used in this work will also fit
their data. The R 0 + R 2 T 2 + R 4.5 T 4.5 fit, which has the same number of free
parameters, is preferable since there seems to be some universality in R 2 . The
low temperature zero field resistivity of La 0.67 Ba 0.33 MnO 3 films [109] has been
fit to R 0 + R 1 T + R 2 T 2 , where R 1 < 0. R 1 may not have physical meaning even
though a small correction is necessary to explain the almost constant
resistivity at the lowest temperatures. A dominant T 2 term has been also
recently found in the resistivity of single crystals [113].

Intrinsic Properties of La 0.67 Ca 0.33 MnO 3
4. 2. 1. 1 Temperature independent term
The temperature independent term in the resistivity can be ascribed to
scattering processes such as impurity, defect, grain boundary and domain wall
scattering. From this value, one can make a rough estimate of the (0 K) mean
free path of 26Å [56]. The temperature independent term in the
polycrystalline samples is somewhat larger than that observed in films or
crystals (Figure 4-6, [29]). This indicates a significant grain boundary resistance
and/or a significantly restricted conduction path in the polycrystalline
samples. The temperature independent terms for the various
La 0.67 Ca 0.33 MnO 3 , La 0.67 Sr 0.33 MnO 3 films and crystals (Table 4-2) concur with
recent measurements on other well annealed La 0.67 AE 0.33 MnO 3 [109, 114, 115]
films and single crystals [116]. This limiting resistivity which is largely in-
dependent of A-site atoms may be due to the inherent disorder on the A-site.
Table 4-2 Magnetoresistance of Annealed Films.
Sample TC (K)
TMR (K)
TMI (K)
R 0
(10 -3 Ωcm)
R2 /R0 (10 -6 K -2 )
R4.5 (0T)/R0 (10 -12 K -4.5 )
R4.5 (7T)/R0 (10 -12 K -4.5 )
Ea /kB (K)
LSM1[a] 360 365 455 0.15 55 28 15
LCM X 267 272 275 0.10 102 67 54 849
LCM13 240 250 250 0.10 61 122 87 967
LCM10 260 275 280 0.12 66 85 72 740
LCM15 260 265 265 0.15 73 87 73 725
LCM17 260 262 264 0.16 60 133 82 865
LCM21 240 245 245 0.28 61 114 60 1116
LBM[109] 310 310 330 0.34 52
4.2.1.2 T 2 dependent term
The T 2 term in the resistivity is also universal with respect to different
samples and type of alkaline earth element (Table 2 ). Furthermore, this term
is independent (within 5%) of magnetic field. The significant
[a] LSM = La .67 Sr .33 MnO 3 ; LCM and LBM are the Ca and Ba analogues. X
refers to float zone grown crystals.
89

90 Chapter 4
magnetoresistance observed in the temperature range used to fit the
conductivity data is absorbed primarily in the T 4.5 term. This universality
with respect to growth, composition, and magnetic field leads us to conclude
that the T 2 term in the resistivity represents intrinsic behavior of
La 0.67 AE 0.33 MnO 3 compounds.
Resistivity (10 -3 Ω cm)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
La 0.67 (Sr/Ca) 0.33 MnO 3 Low Temperature Resistivity
LSM Film
LCM Film
0.1
0.0 0.1 0.2 0.3
(T/T )
C
0.4 0.5
2
Figure 4-8 Low temperature resistivity (in zero field) of
La 0.67 (Sr/Ca) 0.33 MnO 3 films (LSM1 and LCM10). Solid lines are
the fit to R 0 + R 2 T 2 + R 4.5 T 4.5 up to 250K and 200K for LSM and
LCM respectively. The dashed lines show the constant and T 2
terms of the best fit.
Significant T 2 dependencies in resistivity [117] are often observed in
metallic ferromagnets such as Fe, Co, and Ni where the coefficient [118] is 10 -11
Ωcm/K 2 . The magnitude of R 2 is expected [84, 119] to scale with 1/M 0 , which
explains the larger 5 x 10 -8 Ωcm/K 2 value for R 2 found in the weak itinerant

Intrinsic Properties of La 0.67 Ca 0.33 MnO 3
electron ferromagnets Sc 3 In and ZrZn 2 . The term in La 0.67 AE 0.33 MnO 3 (Table
4-2) is about 1000 times larger than that in Fe, Co and Ni, which have
comparable M 0 's. Furthermore, the value of R 2 is expected to decrease in a
magnetic field (proportional to H -1/3 ) due to the suppression of spin
fluctuations; however, the field dependence of R 2 in the manganites is too
small for us to detect. Thus the theory of spin fluctuations does not
completely explain the observed T 2 dependence of the resistivity.
General electron-electron scattering mechanisms within the Fermi liquid
model give a T 2 dependence of the resistivity which is not necessarily field
dependent. A value of R 2 as large as that seen in this work has been found in
the nonmagnetic semimetal TiS 2 [120].
4. 2. 1. 3 Relationship to magnetism
A quite interesting correlation [114, 115, 121, 122] has been found between
the magnetization and the resistivity, namely ρ = R e exp(-M(T, H)/M r ) where
R e and M r are fitting parameters. This is clearly preferable to a polynomial fit
in the critical region near T C . If, at low temperatures, M = M 0 (1-M 2 T 2 ) where
M 2 is largely independent of field, then this expression predicts the resistivity
will vary as R 2 T 2 + R 4 T 4 . Thus a three parameter fit to ρ =
R e exp(-M(T, H)/M r ) is essentially equivalent to R 0 + R 2 T 2 + R 4 T 4 and
therefore fits our data very well. However, this equivalency breaks down
when one considers the field dependence: Since both R e and M r are
independent of field, the field dependence of ρ comes from the field
dependence of M 2 which would affect R 2 and R 4 equally. In summary, the
exponential fit combined with a T 2 dependence of the magnetization uses
four free parameters: R 0 , R e , M r and M 2 (7T) which does not fit the low
temperature data as well as the four free parameters R 0 , R 2 , R 4.5 (0T), R 4.5 (7T)
used in this work. This is discussed further in chapter 7.
91

92 Chapter 4
4. 2. 2 High Temperature resistivity
Above T C , The zero field resistivity (Figure 4-9) of La 0.67 Ca 0.33 MnO 3 fit best
to R h Texp(E a /k b T) predicted by small polaron hopping conductivity [123], as
opposed to that often used for a semiconductor, ln(ρ) ∝ 1/T (Figure 9). Other
(low temperature) forms predicted by variable range hopping [124] or band
semiconductor type conductivity were also fit with two parameters to the data
with less accurate and less consistent results. In contrast to the Ca and Ba
analogs [29, 104] the La 0.67 Sr 0.33 MnO 3 resistivity (Figure 4-10) does not reach a
maximum until about 100K above T C [16].
ln(ρ(10 -3 Ωcm)/T(K))
-3.0
-3.5
-4.0
-4.5
-5.0
-5.5
ln(ρ(10 -3 Ωcm))
2.4
2.2
2.0
1.8
1.6
1.4
1.2
750K
La 0.67 Ca 0.33 MnO 3
Crystal
Film
1.0 1.5 2.0 2.5 3.0 3.5
1000/T (1/K)
750K
a
Crystal
Film
-6.0
0.5 1.0 1.5 2.0
1000/T (1/K)
2.5 3.0 3.5
mW/mg
0.09
0.08
0.07
b
730 740 750 760 770 780
Temperature (K)
Figure 4-9 High temperature resistivity (warming and
cooling) of La 0.67 Ca 0.33 MnO 3 film (LCM17) and crystal in zero
field. Inset a, same data with different abscissa to compare
small polaron and semiconductor models. Inset b, DSC trace
of polycrystalline La 0.67 Ca 0.33 MnO 3 showing the heat of the
high temperature structural transition.

Intrinsic Properties of La 0.67 Ca 0.33 MnO 3
In the theory of small polaron conductivity, E a is the hopping energy. The
prefactor R h ≈ k b /(e 2 a 2 p(1-x)ν) where k b is Boltzmann’s constant, e is the charge
of the hole, a is the distance between hopping sites, p is the carrier density, x is
the doping fraction (1/3 in this case) and ν is the optical phonon or attempt
frequency. A least squares fit of the high temperature data gives a hopping
activation energy of 865K and attempt frequency ν = 6·10 13 /s.
Resistivity (10 -3 Ωcm)
4.0
3.5
3.0
2.5
2.0
T C
ln( ρ(10 -3 Ωcm)/T(K))
-4.6
-4.8
-5.0
-5.2
-5.4
-5.6
-5.8
La 0.67 Sr 0.33 MnO 3 Film
1.0 1.5 2.0 2.5 3.0 3.5
1000/T (1/K)
1.5
300 400 500 600 700 800 900 1000 1100
Temperature (K)
Figure 4-10 High temperature resistivity (warming and
cooling) of La 0.67 Sr 0.33 MnO 3 film (LSM1) in zero field. Inset,
same data displayed as in Figure 4-9.
4. 2. 3 Transport Near T C
The resistance of La 0.67 Ca 0.33 MnO 3 reaches a maximum near the Curie
temperature whereas that of La 0.67 Sr 0.33 MnO 3 maximizes about 100K above T C .
This is best seen from simultaneous magnetization and resistance
93

94 Chapter 4
measurements (Figure 4-6 and Figure 4-7), where there is no error in relative
temperature. Both La 0.67 Ca 0.33 MnO 3 and La 0.67 Sr 0.33 MnO 3 have T MR ≈ T C
within a few Kelvin. For La 0.67 Ca 0.33 MnO 3 T MI is approximately equal to T C ≈
T MR . La 0.67 Sr 0.33 MnO 3 however shows a much more gradual transition to a
hopping conductivity-like transport with T MI (approximately 455K) well above
T C = 360K.
At T C a maximum in dρ(H=0)/dT is expected for a metallic ferromagnet
[125]. This is observed for both compounds studied within experimental
uncertainty. The added resistance at a ferromagnetic Curie temperature is
due to electron scattering off thermally disordered spins and, particularly in
the case of the manganites, polaron formation. Since a magnetic field can
easily suppress spin fluctuations in the critical region, the resistance
associated with magnetic disorder will be reduced in a magnetic field. In a
good metal such as Fe or SrRuO 3 [126] this normally is a small effect of about a
few percent. It has been shown theoretically that this effect is much larger for
a semimetal (or semiconductor) at T C , particularly within the double
exchange model [8, 125, 127, 128]; however, it has been recently been pointed
out that double exchange alone can not account for the large
magnetoresistance [10, 11]. Nevertheless, such an explanation has been used
to explain the magneto-transport properties of semiconducting magnetic
chalcogenides [129] such as EuO [130], Gd 2 S 3 [131], and various spinels [129, 132-
135] where the resistance may drop by several orders of magnitude at T C .
4. 2. 4 Hall effect
The Hall effect data at 5 K (Figure 4-11) were analyzed according to the
equation R = R 0 + H·R H + |H|·R MR , where R H is the Hall resistance and R MR is
the magnetoresistance. The contribution due to the anomalous Hall effect
was not detected at this temperature. From the simple single band interpreta-

Resistance (Ω)
2.88
2.86
2.84
2.82
2.80
Intrinsic Properties of La 0.67 Ca 0.33 MnO 3
La 0.67 (Ca/Sr) 0.33 MnO 3 Hall Effect at 5K
La/Ca Film
La/Sr Film
Hall Effect
-60 -40 -20 0 20 40 60
Field (kOe)
tion of the Hall effect [56], our measurements at 5 K show hole conductivity
with concentrations of the expected order of magnitude (Table 1).
Because the Hall effect is so small and conduction appears to proceed via
small polaron hopping, it is concluded that the temperature dependence of
the resistivity is due to the temperature dependence of the mobility while the
carrier concentration remains relatively constant.
Since LaMnO 3 is a high spin d 4 insulator, the e g orbitals must be split,
possibly by the Jahn-Teller effect. Thus doping with an alkaline earth
element on the La site, should put mobile holes in the lower e g state. The
magnitude of the carrier concentration calculated from the Hall effect is
2.02
2.01
2.00
1.99
1.98
Figure 4-11 Resistance as a function of field
La 0.67 (Ca/Sr) 0.33 MnO 3 films (LSM1 and LCM19) in the Hall
effect configuration at 5 K. The Hall effect is calculated from
the slope of the line indicated (see text).
95

96 Chapter 4
somewhat too large compared with the above model and therefore such a
simple interpretation is clearly inadequate.
4. 2. 5 Crystallographic Phase change
At 750K, a nearly discontinuous jump as well as a significant change in
slope is observed in the resistivity. The jump is much more noticeable in the
crystals indicating a structural transition which is suppressed to some extent
by the epitaxy in the thin films. This transition has been confirmed by
differential scanning calorimetry (Figure 4-9). An orthorhombic to rhombo-
hedral transition is seen in at low temperatures in La 0.83 Sr 0.17 MnO 3 [126].
A change in slope is only noticeable in the ln(ρ) vs. 1/T plot but not in the
ln(ρ/T) vs. 1/T plot. This indicates that the change in dρ/dT can be
interpreted within the small polaron hopping conductivity model [62]. Below
750K thermally activated hopping of the charge carriers dominates the
conductivity, while above 750K scattering of these carriers begins to dominate.
4. 2. 6 Small Polarons
The presence of small polarons implies the existence of local lattice
distortions which localize the charge carriers. A Jahn-Teller distortion about
the Mn 3+ ion is expected from its d 4 configuration in an octahedral
environment. It has been postulated that such a distortion is necessary to
explain the magnetoresistivity of these compounds [10, 11].
Above TC the resistivity of La0.67Sr0.33MnO3 may be explained by a
crossover between two types of polaron conduction [116]. Just above TC the
polarons are heavy free carriers scattered by phonons, which gives a positive
dρ/dT. Quantitatively, this is expected to add a term to the conductivity
proportional to exp(-T/Θ D )/T [62]. At T MI , which should be about half the
Debye temperature (Θ D ), the phonon scattering becomes so strong that the
mean free path is about one lattice spacing, and therefore localizes the

Intrinsic Properties of La 0.67 Ca 0.33 MnO 3
carriers. Above T MI the polarons conduct via thermally activated hopping,
which has a negative dρ/dT. Although this qualitatively explains the data,
the quantitative predication of the temperature dependence does not fit the
data very well.
4. 2. 7 Colossal Magnetoresistance
In general, manganite films reported in the literature which exhibit CMR
as shown in Figure 4-12 [21, 23, 26, 127, 128]: 1) do not have square hysteresis
loops at low temperatures, leading some to believe superparamagnetic
behavior exists in those materials; 2) have ill-defined T C 's which are much
less than those reported for the bulk materials with the same nominal
composition; 3) have T MI well below T C , with T MR slightly less than T MI ; and
4) have high resistivities in the metallic state, at low temperatures.
Figure 4-12 Colossal magnetoresistive La 0.67 Ca 0.33 MnO 3 film
from [21, 127].
The annealed films in this study (which do not exhibit CMR) are much
like bulk polycrystalline or single crystal material: They are clearly itinerant
97

98 Chapter 4
Magnetization
8
6
4
2
La 0.67 Sr 0.33 MnO 3 Polycrystalline Pellet
Magnetization
0
0 2 4 6 8 10
Field (kOe)
electron ferromagnets with reasonably square hysteresis loops, T 2 dependence
of the saturation magnetization and sharp T C 's. They have T C and T MR all
approximately equal and along with T MI are comparable to those of the bulk
polycrystalline material. Finally, the films have low resistivities like those
observed in single crystals [104, 109, 114, 129-131]. It is therefore concluded
that these films are displaying properties inherent to La 0.67 (Sr/Ca) 0.33 MnO 3 and
less influenced by microstructure, strain, and/or compositional variations
than films with suppressed T C and CMR.
Resistivity (10 -3 Ω cm)
0.1248
0.1246
0.1244
0.1242
0.1240
0.1238
Resistivity
La 0.67 Ca 0.33 MnO 3 Film
0.1236
0 10 20 30 40 50 60 70
Field (kOe)
If one assumes that the resistivity behaves as Texp(E a /k B T) above T MI and at
least R 2 T 2 in an infinite field below T C , then the maximum magnetoresistance
0.68
0.64
0.60
0.56
0.52
0.48
Figure 4-13 Simultaneous magnetization and resistivity in a
magnetic field of La 0.67 Sr 0.33 MnO 3 polycrystalline pellet at 5 K.
Data for both increasing and decreasing field are shown. Inset,
Magnetoresistance of La 0.67 Ca 0.33 MnO 3 film (LCM10) at 5 K.
Resistivity (10 -3 Ωcm)

Intrinsic Properties of La 0.67 Ca 0.33 MnO 3
(∆R/R(H) at T = T MR = T MI ) follows exp(E a /k B T MI )/(T MI ). This predicts
"colossal" magneto resistance only at low temperatures, as a consequence of
the depressed T MI . Such a strong dependence of the magnetoresistance on a
film's particular T MI has been documented (Appendices A and B) from the
various values reported in the literature.
4.2.8 Domain Boundary Magnetoresistance
The negative magnetoresistance associated with the domain orientation is
only observed in polycrystalline samples [28, 96]. If this is due to domain wall
scattering, then films and crystals with fewer domain boundaries would be
expected to show a much smaller effect. It has been concluded that spin-
polarized inter-grain tunneling is responsible for this effect [132].
4. 2. 9 Low temperature magnetoresistance
The low temperature negative magnetoresistance linear in H is very small
in the films but somewhat larger in the polycrystalline samples (Figure 4-13,
and [28, 96]). This negative magnetoresistance may be due to the M 2
dependence of the resistivity described in section 7.2.4 [129, 130] combined
with a small forced magnetization. Classic magnetoresistance of metals is
generally positive but is often negative in ferromagnets.
4.3 Conclusion
Well annealed MOCVD thin films show properties of bulk
La 0.67 Ca 0.33 MnO 3 and La 0.67 Sr 0.33 MnO 3 . The limiting low temperature
resistivity of 0.2 mΩcm, which gives a mean free path of roughly 20Å, is
independent of alkaline earth element and only slightly dependent on field
for single crystal material (bulk or film). The manganites show a significant
T 2 dependence of the resistivity which is also independent of alkaline earth
element and magnetic field. The maximum in the magnetoresistance occurs
at the Curie temperature, which is not necessarily where dρ/dT changes sign,
but where dρ/dT is a maximum. A large magnetoresistance can occur well
99

100 Chapter 4
above room temperature, but the effect decreases significantly with
temperature.
Three regions of magnetoresistance are identified in these materials. The
largest effect is likely due to the suppression of magnetic critical scattering
near the Curie temperature. There is also a negative magnetoresistance
associated with the net magnetization of polycrystalline samples but not seen
in single crystal materials. Finally there is a small negative
magnetoresistance linear in field even at low temperatures.
The compounds La 0.67 Ca 0.33 MnO 3 and La 0.67 Sr 0.33 MnO 3 are metallic
ferromagnets with large saturation moments. The saturation magnetization
is found to decrease proportional to T 2 .
The high temperature resistivity of La 0.67 Ca 0.33 MnO 3 clearly follows the law
predicted by small polaron hopping conductivity both in the thermally
activated regime and at higher temperatures where scattering becomes
important. There appears to be a structural transition at about 750K.
The resistivity of La 0.67 Sr 0.33 MnO 3 above the Curie temperature shows a
crossover from a metallic to a hopping regime at higher temperatures.
In conclusion, the CMR effects in materials of the same compositions as
those studied here but with much lower transition temperatures, are not
intrinsic to the thermodynamically stable phase. Local inhomogeneities or
noncrystallinity in not fully annealed films may suppress the magnetic and
metal-insulator transitions. This presumably causes the observed
superparamagnetic type magnetism and conduction via percolation.

5. Local Structure, Transport and Rare Earth Magnetism in
the Ferrimagnetic Perovskite Gd0.67Ca0.33MnO3 In order to examine the nature of the metal-insulator transition in the
CMR manganites, the distorted perovskite Gd 0.67 Ca 0.33 MnO 3 related to
La 0.67 Ca 0.33 MnO 3 was studied. The composition x = 1/3 was chosen because
this doping concentration should maximize the double exchange effect as
seen in the lanthanum compounds [29]. Due to the small size of the Gd 3+
ion, Gd 0.67 Ca 0.33 MnO 3 should be in the region of the x = 1/3 phase diagram
where there is no transition to a metallic state, but to a ferromagnetic
insulating state around 50K [100]. The ferromagnetic nature of this low
temperature state has recently been questioned due to evidence of a spin glass
behavior with no long range ferromagnetic order in (Tb x La 1-x ) 0.67 Ca 0.33 MnO 3
[136]. The x-ray-absorption fine-structure (XAFS) technique can detect small
variations of the average local environment about a particular atomic species,
making it ideal for studying subtle structural phase transitions associated with
any metal-insulator or ferromagnetic transitions. The Mn K-edge (6.54 keV)
and Gd L III -edge (7.25 keV) are well enough separated that there is little
interference of the Mn and Gd XAFS.
In this study, it is found that the rare earth moments in Gd 0.67 Ca 0.33 MnO 3
order antiparallel to the manganese giving rise to ferrimagnetism.
Furthermore, not only does this material remain insulating when it becomes
magnetically ordered, but no structural difference exists between the
ferrimagnetic and paramagnetic state. This is consistent with models that
require a structural change, as well as the ferromagnetic ordering of the
manganese atoms, to explain the metal-insulator transition and the large
magnetoresistance that accompanies it. Most of the work presented in this
chapter has been previously published [137].
101

102 Chapter 5
5.1 Ferrimagnetism
The polycrystalline pellets, crystals and thin films all display a
ferromagnetic transition and a compensation temperature (Figure 5-1), which
is characteristic of ferrimagnets. Both bulk materials, polycrystalline pellet
and float zone crystal, have very similar magnetic properties while the film is
slightly different.
Gd 0.67 Ca 0.33 MnO 3 displays the properties of a ferrimagnet, much like the
garnet ferrite Gd 3 Fe 5 O 3 [138]. At about 60K, the transition metal spins order,
dominating the magnetization. As the temperature is lowered, the opposing
magnetization of the weaker coupled rare earth ions increases, since they
behave almost as though they were free paramagnetic spins reacting to an
internal field (section 5.1.3, Figure 3-19).
At the compensation temperature, T Comp , the rare earth and transition
metal sublattice magnetizations exactly cancel (in zero field). Below T Comp , the
magnetization of the rare earth ions, which have larger spins, dominates.
This is best seen in zero field remnant data which shows the magnetization
changing sign at T Comp .
In the presence of magnetic fields greater than the coercive field, the
stronger sublattice flips to align parallel with the field, giving the minimum
in M at T Comp as seen the 5 kOe data.

Magnetization (µ B /mole)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
Local Structure and Magnetism in Gd 0.67 Ca 0.33 MnO 3
Gd 0.67 Ca 0.33 MnO 3 Magnetization
5kOe
Crystal
Pellet
Film
Remnant
× 25
0 10 20 30 40 50 60 70 80 90
Temperature (K)
Figure 5-1 Low temperature magnetization of
Gd 0.67 Ca 0.33 MnO 3 measured in a 5 kOe field and zero field after
cooling in a large field (remnant).
5. 1. 1 Low temperature moment
The saturation magnetization per mole of Gd 0.67 Ca 0.33 MnO 3 expected from
the Gd 3+ S = 7/2, l = 0 is µ=(0.67 × 2 × 7/2) µ B = 4.67µ B , while high spin
manganese gives spin only (orbital contribution quenched) µ = gSµ B , g = 2 so µ
= 2µ B [0.67 × 2 (from Mn 3+ ) + 0.33 × 3/2 (from Mn 4+ )] = 3.67µ B . The simple two
sublattice (Gd and Mn) ferrimagnetism described above predicts a zero
temperature saturation magnetization of 4.67 µ B - 3.67 µ B = 1.00 µ B (if there is
any canting of the sublattices such as that in Pr 0.67 Ca 0.33 MnO 3 [139], a smaller
moment is predicted). This is roughly what is observed. The mean field
103

104 Chapter 5
calculation predicts a constant moment for T < 5 K, while the data show an
almost linear change in M for 2 K < T < 5 K. Also, the peak in the
magnetization, observed at about 50 K in the 5 kOe data is less than the 1.9µ B
predicted in the mean field calculation.
5. 1. 2 High temperature susceptibility
Susceptibility data (Figure 5-2) above 200 K fit to the Curie-Weiss law:
2 2 2
χ = µ eff /(8(T-Θ)) with Θ = 34 K and µeff = 71 µB . The difference in the
susceptibility of the crystal compared to that of the pellet at high temperatures
2
may not be significant. For quantum mechanical spins µ eff per mole is
expected to be 63 µ B 2 : 2 2 µB 2 (0.67 × 4/2 × 6/2) = 16 µB 2 from Mn 3+ ; 2 2 µB 2 (0.33 ×
3/2 × 5/2) = 5 µ B 2 from Mn 4+ ; 2 2 µB 2 (0.67 × 7/2 × 9/2) = 42 µB 2 from Gd 3+ . This
extra moment, which is also observed in La 0.67 Ca 0.33 MnO 3 [103] section 4.1,
may be due to the orbital contribution in the manganese ions. Since the two
sublattices have different coupling energies, the susceptibility above T C will
not obey a simple Curie-Weiss law even in the mean field approximation
(Figure 5-5). The simple mean field model of the above ferrimagnetism
requires T C /Θ > 3, as described in section 3.2.2.4.1. This is clearly not the case
for the measured parameters. Alternatively, the addition of a Pauli-like
2
susceptibility χ0 to the fit, χ = χ0 + µ eff /(8(T-Θ)), will reduce the contribution
from µ eff . For example, an equally good fit to the data can be found with
2 2
µ eff = 63 µB , χ0 = 7.7 × 10 -6 emu G -1 g -1 , and Θ = 50 K.

1/χ mol (Gauss·mol/emu)
Local Structure and Magnetism in Gd 0.67 Ca 0.33 MnO 3
40
35
30
25
20
15
10
5
Gd 0.67 Ca 0.33 MnO 3 Susceptibility
Pellet
Crystal
5 kOe
0
0 50 100 150 200 250 300 350 400
Temperature (K)
Figure 5-2 Inverse magnetic susceptibility of bulk
Gd0.67Ca0.33MnO3 . Solid line is the high temperature fit to χ =
2
µ eff /(8(T-Θ)) described in the text.
5. 1. 3 Low temperature susceptibility
Between T Comp and T C , the magnetic susceptibility essentially that of the
paramagnetic-like rare earth moments, showing an approximate
χ ∝ 1/(T + T Comp ). Below T Comp the Gd 3+ moments effectively order antiferro-
magnetically with respect to the manganese moments, and so the
susceptibility decreases below T Comp .
105

106 Chapter 5
χ mol (µ B /mol/T)
0.40
0.36
0.32
0.28
0.24
0.20
High Field Susceptibility of Gd 0.67 Ca 0.33 MnO 3
Magnetization (µ B /mole)
0 10 20 30 40 50 60 70 80
Temperature (K)
3
2
1
0
5K
0 10 20 30 40 50 60 70
Magnetic Field (kOe)
Figure 5-3 Low temperature and high-field magnetic
susceptibility, χ = (M(60 kOe)-M(40 kOe))/20kOe, of
Gd 0.67 Ca 0.33 MnO 3 crystal. Inset, hysteresis loop at 5 K.
The hysteresis loop at 5 K (Figure 5-3) shows an additional high field
paramagnetic response not expected for a mean field ferrimagnet. It is not
uncommon, however, to observe such a field-induced increase of the
magnetization in canted ferromagnetic or ferrimagnetic systems [140, 141],
and is used as evidence of the ferrimagnetic sublattices canting away from
antiparallel [101]. The magnetic field affects the canting angle, which results
in the magnetization increasing nearly linearly with field even at T = 0 K
[140]. If this large susceptibility were due to a second phase with free
paramagnetic spins, then this phase would have to contain at least half of the
available spins and have a 1/T temperature dependence for all T. This is not

Magnetization 2 (µ B /mole) 2
0.10
0.08
0.06
0.04
0.02
Local Structure and Magnetism in Gd 0.67 Ca 0.33 MnO 3
Gd 0.67 Ca 0.33 MnO 3 Arrott Plot
35K
40K
45K
50K
55K
60K
70K
0.00
0 2000 4000 6000 8000 10000
H/M (Oe/µ )
B
Figure 5-4 Arrott plot of polycrystalline Gd 0.67 Ca 0.33 MnO 3
pellet.
seen in Figure 5-3. Instead, the high field susceptibility maximizes near T Comp
which is typical of an antiferromagnet with T N ≈ T Comp . This is predicted in the
mean field calculation for a ferrimagnet described in section 3.2.2.4.1.
5. 1. 4 Near T C magnetism
The experimental values of T C are summarized in Table 5-1. An example
Table 5-1 Transition Temperatures for Gd 0.67 Ca 0.33 MnO 3 .
T Comp Arrott T C Remnant T C Inflection T C
Polycrystalline Pellet 17K 52K 60K 80K
Single Crystal 17K 56K 62K 78K
Thin Film 13K 70K 81K 88K
107

108 Chapter 5
of the nonlinear M 2 vs. H/M Arrott plot is given in Figure 5-4.
5. 1. 5 Magnetism model
To explain the magnetism, a mean field model was used assuming a
ferrimagnetic ground state. Other possible explanations for the magnetism in
Gd 0.67 Ca 0.33 MnO 3 are briefly discussed
5 kOe
Gd 0.67 Ca 0.33 MnO 3 Mean Field Calculation
Remnant
Gd
Mn
20 40 60 80 100 -5
Temperature (K)
3
2
1
0
-1
-2
-3
-4
Magnetization (µ B /mol)
5 kOe
0 50 100 150 200 250
0
300
Temperature (K)
Figure 5-5 Magnetization and inverse magnetic susceptibility
calculated for Gd 0.67 Ca 0.33 MnO 3 using the simplified mean field
theory described in the text and T C = 83 K, T Comp = 17 K. The
contribution to the magnetization of each sublattice is shown
in dashed lines.
5. 1. 5. 1 Mean Field Model
Due to the stronger 3d exchange interaction compared to that of the 4f, the
Mn-Mn coupling is expected to be stronger than the Mn-Gd, and the Gd-Gd
interaction to be negligible. Since the observed Curie temperature is about
40
35
30
25
20
15
10
5
1/χ (mol G/emu)

Local Structure and Magnetism in Gd 0.67 Ca 0.33 MnO 3
that expected from the trend found in the manganese ferromagnetic ordering
temperature as a function of the average of the R and A atom sizes in
R 1-x A x MnO 3 [100], it is assumed for now that the Mn-Mn interaction in
Gd 0.67 Ca 0.33 MnO 3 is ferromagnetic. The ferrimagnetism must then arise from
a weaker, antiferromagnetic Mn-Gd interaction. Thus, unlike the ferrites,
where the dominant interaction is antiferromagnetic between the two
transition metal sublattices, the strongest magnetic coupling in
Gd 0.67 Ca 0.33 MnO 3 is ferromagnetic, much like that found in the rare earth
cobaltites, e. g. GdCo 5 [142].
The above description of the magnetization can be bolstered using a
simple molecular or mean field model described in section 3.2.2.4. Assuming
that all atoms on each sublattice (Mn or Gd) feel the same mean field, there
are only two unknown mean field coefficients due to the Mn-Mn and Mn-Gd
interactions. These two coefficients are easily determined since they are
nearly proportional to T C and T Comp , respectively. This molecular field model
(Figure 5-5) reproduces the qualitative features described above.
The calculated magnetization and inverse susceptibility for the molecular
field model described below is shown in Figure 5-5.
This model is limited by the obvious simplifications: not only are there
two distinct types of Mn atoms (Mn 3+ and Mn 4+ ), but there are numerous
possible configurations for the number and type of near neighbors due to the
randomness on the Mn and Gd/Ca sites. The discrepancy among the various
values of T C determined by the three methods mentioned is an example of
non-mean-field ferromagnetic behavior. The discrepancies in the salient
features discussed above may result from the simplifications of the model,
and the assumptions behind it.
5. 1. 5. 2 Canted antiferromagnetism
The evidence concerning the magnetic structure of the manganese in the
ferromagnetic insulator phase points to a canted antiferromagnetic structure
109

110 Chapter 5
rather than a simple ferromagnet. The magnetic structure of Pr 0.67 Ca 0.33 MnO 3
which is on the boundary between ferromagnetic-insulator and ferro-
magnetic-metal low temperature phases [100], has been extensively studied.
Neutron diffraction shows Pr 0.67 Ca 0.33 MnO 3 to have a charge-ordered, canted-
antiferromagnetic structure in the low temperature phase with ferromagnetic
moment about half of the total expected moment [139, 143]. A compound
clearly in the ferromagnetic-insulator part of the tolerance-factor phase-
diagram (Figure 3-17), much closer to Gd 0.67 Ca 0.33 MnO 3 than Pr 0.67 Ca 0.33 MnO 3
is La 0.2 Y 0.5 Ca 0.3 MnO 3 . This compound appears to be ferromagnetic with about
half the expected ferromagnetic moment [100], which could be explained by a
canted antiferromagnetic state similar to that found in Pr 0.67 Ca 0.33 MnO 3 . Thus,
one should expect the ferromagnetic moment of the Mn ions in
Gd 0.67 Ca 0.33 MnO 3 to be half that predicted in the above molecular field model.
If the Gd moments are also canted to the same degree, then the predicted
magnetization will be qualitatively the same for the uncanted case
(Figure 5-5), only half the magnitude; namely, there will still be a
compensation temperature.
5. 1. 5. 3 Spin glass magnetism
Atomic disorder such as that found on both the Gd and Mn site in
Gd 0.67 Ca 0.33 MnO 3 can result in a spin-glass, and indeed a spin-glass insulating
low-temperature phase was found in (Tb x La 1-x ) 0.67 Ca 0.33 MnO 3 in place of the
ferromagnetic insulator state. The compound Tb 0.67 Ca 0.33 MnO 3 should be in
the same region of the tolerance-factor phase-diagram as Gd 0.67 Ca 0.33 MnO 3 and
La 0.2 Y 0.5 Ca 0.3 MnO 3 . The Tb compound is apparently a pure spin-glass with no
long-range magnetic order [136]. A full investigation of the existence or non-
existence of spin-glass behavior in Gd 0.67 Ca 0.33 MnO 3 is beyond the scope of this
work. However, it is clear that Gd 0.67 Ca 0.33 MnO 3 has long range magnetic
order, which eliminates the possibility of a spin glass. The distinction is most
obvious in the Arrott plot (Figure 4), which shows that Gd 0.67 Ca 0.33 MnO 3 has a

ln(ρ/T (Ω cm/K))
Local Structure and Magnetism in Gd 0.67 Ca 0.33 MnO 3
6
4
2
0
-2
-4
-6
ln(ρ/Ω cm)
Gd 0.67 Ca 0.33 MnO 3 High Temp. Resistivity
4
3
2
1
1.0 1.5 2.0 2.5 3.0
1000/T (1000/K)
1 2 3 4 5 6 7
1000/T (1000/K)
non-zero magnetization when H = 0 below T C , while
(Tb 0.33 La 0.67 ) 0.67 Ca 0.33 MnO 3 does not [136]. The other striking feature of
Gd 0.67 Ca 0.33 MnO 3 is the existence of a compensation point, which to our
knowledge cannot be explained by a spin-glass magnetic system.
5.1.5.4 Related Compounds
Since Gd 3+ has the largest spin of the rare earth elements, it should have
the highest ordering temperature. If the relevant exchange interaction is due
to spin-spin coupling of the rare earth and manganese atoms, then the
strength of this interaction will be proportional to the spin S of the rare earth
a
ln(ρ/Ω cm)
4
3
2
1
b
0.18 0.20 0.22
1/T 1/4 (K -1/4 )
Figure 5-6 High temperature resistivity during heating and
cooling a Gd 0.67 Ca 0.33 MnO 3 film, ln(ρ/T) vs. 1/T. Inset a,
comparison with ln(ρ) vs. 1/T. Inset b, comparison with ln(ρ)
vs. 1/T 1/4 .
111

112 Chapter 5
ln(ρ /Ω cm)
24
22
20
18
16
14
12
10
Gd 0.67 Ca 0.33 MnO 3 Low Temperature Resistivity
ln(ρ/T (Ω cm/K))
20
15
10
ion. From this relation the ordering temperature of the rare earth moments
in related compounds is estimated to be about 7 K for Nd 0.67 Ca 0.33 MnO 3 and
5 K for Pr 0.67 Ca 0.33 MnO 3 . This should result in a decrease in M at low T. This
appears at temperatures lower than these estimates, if at all [23, 144]; perhaps
indicating an interaction strength proportional to S 2 , the square of the rare
earth spin, or simply that these moments do not order antiparallel to the
manganese moments [145].
5
10 12 14 16 18 20 22
1000/T (1/K)
65 K ≈ Tc
0.32 0.33 0.34 0.35 0.36 0.37 0.38
a
0.0
-0.2
-0.4
-0.6
1/T 1/4 (K -1/4 )
300K MR (%)
b
300K
200K
200K MR (%)
-1
-2
-0.8
-3
-60 -40 -20 0 20 40 60
Field (kOe)
Figure 5-7 Low temperature resistivity of Gd 0.67 Ca 0.33 MnO 3
crystal, ln(ρ) vs. 1/T 1/4 . Inset a, comparison with ln(ρ/T) vs.
1/T. Solid lines show linear best fit to the data shown. Inset b,
magnetoresistance of a film at 200 K and 300 K; solid line is
the quadratic fit.
0

Local Structure and Magnetism in Gd 0.67 Ca 0.33 MnO 3
5.2 Electronic Transport
The resistivity can be well approximated by R h Texp(E a /k B T) (Figure 5-6),
predicted by small polaron hopping conductivity [68, 146]. Below about 150 K
(Figure 5-7), the resistivity is somewhat better described by R 0 exp(T 0 /T) 1/4 ,
which is commonly attributed to variable range hopping [123]. The
quantitative aspects of the fits is discussed below. The thin film and crystal
samples show no transition to a metallic state down to 5 K.
5.2.1 Magnetoresistance
No extraordinary magnetoresistance was observed in the entire
temperature range. The magnetoresistance at 200 K and 300 K is a few percent
in a 70 kOe field and approximately proportional to H 2 (Figure 5-7). Reliable
resistivity data could be measured on the crystal sample down to 40 K. No
discontinuity in the resistivity was found near T C .
5. 2. 2 Small Polaron Hopping
The electrical resistivity of Gd 0.67 Ca 0.33 MnO 3 is consistent with adiabatic
small polaron hopping conductivity (section 4.2.6)R h Texp(E a /k B T) [58, 68]
R h = 2k B /3ne 2 a 2 ν particularly at high temperatures. Here, k B is Boltzmann’s
constant, e is the electronic charge, n is the number density of charge carriers
— about 0.7 carrier per Mn, a is the site-to-site hopping distance, and ν is the
longitudinal optical phonon frequency. This behavior has also been observed
in the La containing compound described in section 4.2.2 [103, 146].
Resistivity data are often displayed on an Arrhenius plot (ln(ρ) vs. 1/T) which
will give a straight line for a band gap semiconductor [15, 147]. The slightly
different temperature dependence of the data provides the significant
nonlinearity in such a fit (Figure 5-6). A polaron hopping activation energy,
E a , of 1940K, and an attempt frequency of 8 × 10 13 s -1 is inferred. The small
polaron mechanism implies that unlike a bandgap semiconductor, the highly
temperature dependent quantity is the mobility, not the carrier concentration.
113

114 Chapter 5
A high, temperature independent carrier concentration is consistent with
Hall effect measurements (sect 4.2.4) on La 0.67 Ca 0.33 MnO 3 .
5. 2. 3 Variable Range Hopping
The resistivity of the manganites has also been attributed to variable range
hopping due to Anderson localization [148]. At temperatures less than 150K,
this picture produces a somewhat better fit to our data (Figure 5-7), but is
clearly inferior to the polaron model at high temperatures (Figure 5-6). The
fit to R 0 exp(T 0 /T) 1/4 gives T 0 = 1.7 × 10 9 K. In the theory of variable range
hopping [71, 149], k B T 0 ≈ 21/(ζ 3 N(E F )) where ζ is the decay length of the
localized wave function, and N(E F ) is the density of localized states at the
Fermi level. For ζ ≈ a = 3.9Å, the distance between neighboring Mn atoms,
this implies N(E F ) ≈ 2.4 × 10 18 eV -1 cm -3 which is typical for disorder
semiconductors but about 1500 times less than the density of states found in
the specific heat of manganites which become metallic at low temperatures
(≈ 5 mJ mol -1 K -2 )[124].
5.3 X-ray Absorption Fine Structure
The primary quantity in XAFS data analyses is χ(k)=(µ(k)-µ 0 (k))/µ 0 (k)),
where k is the ejected photoelectron wave vector, µ(k) is the total absorption
due to the absorbing atomic species, and µ 0 (k)) is the portion of µ(k) that does
not include the photoelectron backscattering off neighboring atoms. This
backscattering causes an interference at the absorbing atom which is manifest
as oscillations in χ(k). A Fourier transform of kχ(k) thus produces peaks that
correspond to the distribution of atoms around the absorbing atom.
Figure 5-8 shows these Fourier transforms.

Local Structure and Magnetism in Gd 0.67 Ca 0.33 MnO 3
Figure 5-8 Fourier transform of kχ(k) from (a) Mn K-edge and
(b) Gd L III -edge data on Gd 0.67 Ca 0.33 MnO 3 . The solid lines are
data collected at T = 69 K, while the triangles (∆) are data
collected at T = 40 K. Agreement between data above and
below T C is well within the errors of the experiment.
Transform ranges for the Gd edge data are from 3.5-12.5 Å -1
and Gaussian broadened by 0.3 Å -1 . Transform ranges for the
Mn edge data are from 3.2-12.5 Å -1 and Gaussian broadened by
0.3 Å -1 .
The agreement between the data above and below T C is very good and
places limits on any atom-position changes around either the Gd or M n
atoms within ±0.005 Å, and changes in the Debye-Waller broadening factor to
±0.002 Å (Pnma; a = 5.52Å, b = 7.50Å, c = 5.34Å).
115

116 Chapter 5
5. 3. 1 Relationship of structure to CMR
Unlike the colossal magnetoresistance manganites, Gd 0.67 Ca 0.33 MnO 3 does
not have a metal-insulator transition or a large magnetoresistance near the
Curie temperature. The negative magnetoresistance is only of order 1% in
large fields and is proportional to H 2 (or M 2 since it scales with χ 2 ) like that
observed in La 0.67 Ca 0.33 MnO 3 in the paramagnetic state (section 7.2.1) [150].
Thus, Gd 0.67 Ca 0.33 MnO 3 is clearly in the class of 1/3 doped manganites which,
due to the small size of Gd 3+ [100], change from paramagnetic insulators to
ferromagnetic insulators at T C . There is much interest in finding a structural
change at T C in the giant magnetoresistive manganites, since it is predicted to
have a large effect on the resistive transition [11]. There is now evidence [151-
154] for a decrease in structural disorder as the GMR manganites become
ferromagnetic metals. Gd 0.67 Ca 0.33 MnO 3 shows the contrapositive: there is no
discontinuity in the atomic disorder for a material which does not become
metallic at T C . This has also been noted in manganites with different dopant
concentrations: La 0.88 Ca 0.12 MnO 3 [151] and La 0.5 Ca 0.5 MnO 3 [154].
5.4 Conclusion
Gd 0.67 Ca 0.33 MnO 3 is ferrimagnetic with a compensation temperature of
about 15K due to the interaction and ordering of the Gd 3+ ions. The
qualitative features of the magnetic properties can be accounted for with a
simple two sublattice (Gd and Mn) molecular field model. The large high
field susceptibility at low temperatures may indicate a canting of the magnetic
sublattices. Contrary to the conclusion in [136] that there does not exist long-
range magnetic order (only spin-glass magnetism) in the ferromagnetic-
insulator region of the tolerance-factor phase-diagram, Gd0.67Ca0.33MnO3 shows definite long-range order and a compensation point which can not be
explained by spin-glass phenomena alone. The resistivity is consistent with
small polaron conductivity over a broad temperature range, with a possible
crossover to a different mechanism such as variable-range hopping at low

Local Structure and Magnetism in Gd 0.67 Ca 0.33 MnO 3
temperatures. There is no noticeable change in the structure, as determined
by XAFS, or conductivity at the ferrimagnetic transition. Recent theoretical
and experimental work conclude that structural effects at T C are associated
with the metal-insulator transition and large magnetoresistance. Since
Gd 0.67 Ca 0.33 MnO 3 is in the doping regime where a metal-insulator transition
and large magnetoresistance are not observed at T C , this work supports that
conclusion, i. e. that a significant structural change is associated not with the
ferromagnetic transition, but the metal-insulator transition.
117

6. Magnetoconductivity in La 0.67Ca 0.33MnO 3
In this chapter, the nature of the field dependence of the resistivity in
La 0.67 Ca 0.33 MnO 3 - or magnetoresistance above and below T C is studied in more
detail. For this study, sample LCM 17 (Chapter 4, Table 4-2) was used since it
shows magnetic and transport properties consistent with bulk samples.
Measurements at 0.9T C = 237 K and 1.1T C = 289 K will have considerable
magnetoresistance while being clearly either in the ferromagnetic or
paramagnetic state. In order to achieve maximum temperature stability, data
were collected only after there was no appreciable drift in the resistivity. The
major conclusions have been reproduced with other samples which like this
one have high T C ’s and no domain boundary magnetoresistance (section
4.2.8). The work presented in this chapter has been published in [150].
Two configurations were used: 1) magnetic field parallel to the film,
which was used for longitudinal and transverse magnetoresistance analysis
since there is almost no demagnetization field. Since the van der Pauw
configuration is used, the longitudinal magnetoresistance has a small
component due to transverse magnetoresistance and vice versa. 2) magnetic
field perpendicular to the film, used primarily for Hall effect measurements.
6.1 Anisotropic magnetoresistance
The magnetoresistance at 289 K (1.1T C ) on increasing and decreasing
magnetic field is displayed in Figure 6-1 for longitudinal and transverse
directions. The two configurations give identical magnetoresistance.
The high field (|H| > 1 kOe) magnetoresistance at 0.9T C is also shown in
Figure 6-1 for both increasing and decreasing fields. There is no appreciable
hysteresis for |H| > 200 Oe. The transverse and longitudinal
magnetoresistance in high fields are nearly identical when the same current
path is used. Different current paths however, give a slightly different
118

Magnetoconductivity in La 0.67 Ca 0.33 MnO 3
magnetoresistance at 0.9T C indicating some magnetic/electronic
inhomogeneity below T C . All current paths have the functional form of the
magnetoresistance described below, and therefore the apparent
inhomogeneity does not affect the conclusions of this paper. The transverse
magnetoresistance with the field perpendicular to the film is identical to the
transverse magnetoresistance with the field parallel to the film with an
additional demagnetization field of (1750+/-100 Oe); this was used to estimate
M at 0.9 T C .
6.2 Magnetoresistance models
In order to assess the potential of the colossal magnetoresistance in a
magnetic field sensor, (such as the read head in a magnetic recording device)
knowledge of the magnitude and exact field dependence of the
magnetoresistance is critical, primarily in low magnetic fields, H.
Experimental data generally show that for low fields the magnetoresistance is
approximately quadratic in H above the Curie temperature, T C and is more
cusp like or linear below T C [116]. There exists, however, many counter
examples to this generalization [22, 25, 114] presumably due to inhomogenous
samples where T C is not so well defined [116]. The best empirical model of the
resistivity (ρ = ρ 0 e M/M 0) around T C [114] gives a linear field dependence. Since
the earliest reports [4], the magnetoresistance has been claimed or assumed to
be independent of the angle between the magnetic field and the current [113].
The microscopic mechanism of the magnetoresistance can be quite
complicated and still not accurately predict much of the observed
magnetoresistance phenomena. Much of the theory assumes an electron
band transport mechanism [55], which may not be appropriate in the case of
the manganites because they are not metallic above T C . Some microscopic
models of transport in the manganites have predicted the conductivity σ or
119

120 Chapter 6
resistivity ρ ∝ M 2 [113, 155, 156] (which is isotropic) with limited experimental
verification.
6. 2. 1 General Model
The difference above and below T C as well as the relationship to the
magnetization M can be shown more generally by only considering the
symmetry of the resistivity tensor - with no assumptions about or even a
reference to microscopic transport mechanisms. Following Landau and
Lifshitz [157], the relation between the electric field E and the current density J
is given by the resistivity tensor ρ: E i = ρ ik J k where the components of ρ are
functions of H and M. Since H and M are vectors which are antisymmetric
under time reversal, ρ must have the following symmetry: ρ ik (H,M) =
ρ ki (-H,M) = ρ ki (H,-M) = ρ ik (-H,-M). The symmetric, s ik , and antisymmetric, a ik ,
parts of ρ ik = s ik + a ik then have the following properties: s ik (H,M) = s ik (-H,M) =
s ik (H,-M) and a ik (H,M) = -a ik (-H,M) = -a ik (H,-M). Thus the components of s ik are
even functions of H and M while those of a ik are odd functions.
The measured resistivity ρ, the diagonal component of the resistivity
tensor, contains no contribution from a ik and therefore must be even in H and
M. Assuming the resistivity is an analytic function with respect to H and M
(which themselves may not be analytic in T), the first terms in the expansion
of ρ in powers of H and M is then ρ = ρ 0 + αM 2 + βH 2 + γH•M. At this point,
one can already expect that for T > T C (M=0) the low field magnetoresistance
should be proportional to H 2 while for T < T C (M≠0) the term linear in H may
be expected to dominate. Clearly a negative magnetoresistance can not obey
this relation for arbitrarily large H, for the resistivity would eventually
become negative; higher order terms or inverse powers of H are then
required. The above argument is as valid for the conductivity tensor σ, where
J i = σ ik E k , as it is for ρ. In the case of the manganites, the magnetoconductivity

Magnetoconductivity in La 0.67 Ca 0.33 MnO 3
is positive and unlike the magnetoresistance may not require further
expansion in H.
6.2.1.1 Magnetoconductivity model
For an isotropic conductor the longitudinal and transverse
magnetoconductances (and magnetoresistances) are, in general, not equal.
The general form of the relation between J and E in an isotropic conductor
[157] up to terms quadratic in H and M is:
J = σ 0 E + α 1 M 2 E + α 2 (E•M)M +
β 1 H 2 E + β 2 (E•H)H +
γ 1 (H•M)E + γ 2 (M•E)H + γ 3 (E•H)M + (1)
σ H H×E + σ A M×E
If all the coefficients (except α 1 ) are allowed to be functions of M 2 , (e.g.
σ 0 → σ 0 + α 1 M 2 + … ) then (1) is a complete expansion in H up to H 2 and all
powers of M.
The β i terms provide the M = 0 magnetoconductance quadratic in H.
When β 2 = 0, the longitudinal and transverse magnetoconductances are
equal. Similarly, the α 2 term leads to magnetoconductance depending on the
angle between E and M - known as anisotropic magnetoresistance 6.1. The γ i
provide magnetoconductance linear in H only when M ≠ 0. The equation
used to fit the data is formally equivalent to a circuit containing a resistor (ρ ∞ )
in series with the magnetoconductor (e.g. σ(H) = σ 0 + σ Η 2H 2 ), although the
physical significance of such an equivalent circuit is not entirely clear. In any
event, ρ ∞ is presumably the intrinsic spin independent resistivity found to be
proportional to a constant plus a T 2 term, described in sections 4.2.1 and 7.2.2
[103].
121

122 Chapter 6
The Hall effects are contained in the antisymmetric part of σ and ρ and
therefore can be expanded in odd powers of H and M. The first two terms in
the Hall resistivity is R H H×J + R A M×J, where R H is the normal Hall coefficient
and R A is the anomalous Hall coefficient. Since there is no evidence to prefer
the Hall conductivity of equation 1 (≈ -R H /H 2 ) to the Hall resistivity, the Hall
effect is analyzed using R H and R A .
It is possible that the observed magnetoconductivity could be reduced to
terms involving only M 2 and (E•M)M since M ≈ M 0 + χH gives the same field
dependence described above. This would imply a strict relationship among
the various β i (T) and γ i (T) which could be determined by simultaneous
measurements of M and the magnetoconductivity. The resistivity saturates
in a magnetic field [113, 158] more rapidly than does the magnetization
(giving M 2 behavior only at small M), indicating again that it is probably the
conductivity not the resistivity which is proportional to M 2 . This is discussed
in more detail in chapter 7.
The reciprocal nature of the conductivity and resistivity leads to only an
abstract distinction between them. Thus, the simpler, empirical description of
the resistivity in terms of a magnetoconductivity may be accidental.
Nevertheless, it warrants further consideration. The standard additive
scattering time interpretation of the magnetoresistivity requires a more
complex H dependence to describe the suppression of magnetic scattering by a
magnetic field. The simplicity of the magnetoconductivity expression may
signify the basis of an alternative transport interpretation. Taken literally, the
magnetic field appears to open parallel channels of conduction with a simple,
physical functional dependence.

MagnetoResistance ∆R/R(0) (%)
10
0
-10
-20
-30
-40
-50
6.2.1.1.1 T > T C regime
Magnetoconductivity in La 0.67 Ca 0.33 MnO 3
ρ ∞
1/βH 2 or 1/γ|Η|
1/σ 0
La 0.67 Ca 0.33 MnO 3
T = 0.9Tc
T = 1.1Tc
-60
-80 -60 -40 -20 0 20 40 60 80
Magnetic Field (kOe)
Figure 6-1 High field (longitudinal) magnetoresistance above
and below T C for La 0.67 Ca 0.33 MnO 3 film. The solid lines show
the fit using the indicated equivalent circuit
The 1.1T C data fit very well with three parameters (ρ ∞ = 1.3 mΩcm, σ ο =
0.14 (mΩcm) -1 and σ Η 2 = 43 × 10 -6 (mΩcm) -1 (kOe) -2 ) to ρ(H) = ρ ∞ + 1/(σ 0 + βH 2 ).
A much less satisfactory fit is obtained with ρ(H) = ρ(0) + aH 2 + bH 4 .
Above T C , M = 0 (or at least M = χH which gives the same field
dependence of σ) so the diagonal element of the isotropic magneto-
conductance tensor σ(H) = σ 0 + β 1 H 2 + β 2 (E•H) 2 /E 2 . From the fit described
above at 1.1 T C, β 1 = 43 × 10 -6 (mΩcm) -1 (kOe) -2 and β 2 ≈ 0 is estimated.
123

124 Chapter 6
∆R/R (%)
1
0
-1
-2
-3
-4
Transverse
Sum
Longitudinal
-600 -400 -200 0 200 400 600 -1
Field (Oersted)
Magnetization
Figure 6-2 Low field magnetoresistance and magnetization
(relative units) of La 0.67 Ca 0.33 MnO 3 at 0.9 T C . The sum of the
longitudinal and transverse resistances minimizes the effect
of the anisotropic magnetoresistance.
6.2.1.1.2 T < T C regime
4
3
2
1
0
Relative
Magnetization
The longitudinal data fit well with three parameters (ρ ∞ = 0.58 mΩcm, σ ο =
0.35 (mΩcm) -1 and σ Η = 0.013 (mΩcm kOe) -1 ) to ρ(H) = ρ ∞ + 1/(σ ο + σ Η |H|). A
significantly less satisfactory fit is obtained with an exponential form [114,
156]: ρ(H) = ρ ∞ + ae -|H|/b .
Below T C and for M ≠ 0 parallel to H, i. e. M = MH/|H|, then up to terms
linear in H, σ(H) reduces to (σ 0 + α 1 M 2 ) + α 2 M 2 (E•H) 2 /E 2 H 2 + γ 1 M|H| +
(γ 2 + γ 3 )M|H|(E•H) 2 /E 2 H 2 . By fitting the 0.9T C data, γ 1 = σ Η /M = 73 × 10 -6 (mΩcm

Magnetoconductivity in La 0.67 Ca 0.33 MnO 3
kOe emu) -1 is estimated. For equal longitudinal and transverse
magnetoresistance, γ 2 + γ 3 ≈ 0.
The |H| form of the magnetoresistance below T C provides a cusp in the
magnetoresistance near H = 0. This is in contrast to the rounded curve
observed above T C .
6.2.1.1.3 Anisotropic magnetoresistance
The low field data are shown in Figure 6-2 with the average of the
transverse and longitudinal magnetoresistances. The vertical offset of these
data is approximate, i.e. it is calculated from measurements of a different,
patterned sample. The switching of the resistivity at +/-65 Oe is due to the
switching of the magnetization (coercivity). This can be seen in Figure 6-2
where the magnetization, relative to the saturated 0.9 T C value of about
1µ B /Mn, is shown for comparison.
The anisotropic magnetoresistance (AMR) at 0.9T C appears to provide the
nonlinear magnetoresistance in low fields, (Figure 6-2) aside from the
magnetization switching at 65 Oe. Since 1/(ρ trans + ρ long ) ≈ α 2 (E trans •M) 2 +
α 2 (E long •M) 2 ≈ α 2 M 2 (Cos 2 (θ) + Cos 2 (θ + 90°)) = α 2 M 2 , the sum of the transverse
and longitudinal resistances should be independent of the direction of M,
leaving only the magnetoresistance linear in H (Figure 6-2). This is
approximately what happens. The larger (smaller) than linear peak
(depression) in the longitudinal (transverse) magnetoresistance for H < 100
Oe is apparently due to AMR since the sum produces a curve which looks
like |H| that switches at the coercive field. Similar AMR has been
independently verified elsewhere [159].
125

126 Chapter 6
R(H)-R(-H) (µΩ cm)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
La 0.67 Ca 0.33 MnO 3 Hall Effect
T = 1.1 T C
T = 0.9 T C
0.0
0 10 20 30 40 50 60 70
Magnetic Field (kOe)
Figure 6-3 Hall effect of La 0.67 Ca 0.33 MnO 3 below (fully
magnetized data only) and above T C .
6.2.1.1.4 Hall Effect
The Hall effect (Figure 6-3) is calculated from R(H) - R(-H), which should
remove all contributions to the transverse voltage due to magnetoresistance
and changes in magnetic homogeneity. The Hall effect at 0.9T C shows a clear
contribution at H=0 due to the anomalous Hall effect [160] as well as a term
linear in H. From this linear term, a carrier density of n = 0.85 holes/cell is
calculated which is the same as that observed at 5 K (section 4.2.4), where no
anomalous Hall effect was detected. The anomalous Hall effect apparently
increases with temperature, which is quite unexpected since it is usually
proportional to M [160]. Nevertheless, this has also been observed in the
magnetic spinels such as CuCr 2 Se 4 [132-134]. At 1.1 T C the Hall effect has an

Magnetoconductivity in La 0.67 Ca 0.33 MnO 3
electron-like slope at low fields that slowly reverses to hole-like in high fields.
The low field slope can be attributed to the anomalous Hall effect resulting
from the magnetization due to the Curie-Weiss susceptibility, χ: R hall = R A M +
R H H = (χR A + R H )H. Thus while the sample magnetizes the Hall effect looks
electron-like, but once the magnetization saturates, the Hall effect returns to
hole-like.
6.3 Conclusion
In summary, an accurate and simple model, based solely on symmetry
without reference to any mechanism, is proposed for the magnetoresistance
in La 0.67 Ca 0.33 MnO 3 . One can also conclude from this model that the
mechanism of the magnetotransport is best described by a
magnetoconductance, and therefore mechanistic theories should predict at
least a M 2 dependence of the conductivity. The functional form and estimate
of parameters presented above should not only help guide the development
of a mechanistic theory, but also provide a basis for predicting the
magnetotransport properties when designing a device.
In homogeneous materials, it is clear that the magnetoresistance
maximizes near the ferromagnetic Curie temperature T C and rapidly
decreases at lower or higher temperatures. The magnetoresistance also slowly
saturates as the magnetic field is increased past several Tesla. Thus, the
conductivity behaves much like the magnetization: above T C the
conductivity and magnetization are low, while below T C the conductivity and
magnetization rapidly increase. Furthermore, near T C both the conductivity
and resistivity can be increased by an applied magnetic field. The theory of
double exchange, which has been developed to explain these properties of the
manganites [7, 112, 155], predicts this correlation between the conductivity and
the magnetization.
127

7. Critical Transport and Magnetization of La 0.67Ca 0.33MnO 3
In homogeneous La 0.67 Ca 0.33 MnO 3 , it is clear that the magnetoresistance
maximizes near the ferromagnetic Curie temperature T C and rapidly
decreases at lower or higher temperatures. The magnetoresistance also slowly
saturates as the magnetic field is increased past several Tesla. Thus, the
conductivity behaves much like the magnetization: above T C the
conductivity and magnetization are low, while below T C the conductivity and
magnetization rapidly increase. Furthermore, near T C both the conductivity
and magnetic moment can be increased by an applied magnetic field. The
theory of double exchange, which has been developed to explain these
properties of the manganites [7, 112, 155], predicts this correlation between the
conductivity and the magnetization.
The relationship between the conductivity (or resistivity) and the
magnetization has been examined both experimentally and theoretically.
Several theoretical models [31, 112, 121, 155, 156, 161-164] predict an M 2
dependence of the resistivity (or conductivity). The first term of the Taylor
series expansion of the resistivity (or conductivity) in terms of M should be
M 2 for all models due to symmetry considerations [150].
Experimentally, a correlation between the resistivity and magnetization
can be found by plotting one as a function of the other [24, 114, 115, 121, 158,
165]. These plots show that the resistivity is roughly proportional to the
square of the magnetization M 2 for small M. For larger M however, the
resistance decreases more slowly in relationship to M 2 as if it were saturating
before the magnetization does. Thus the relationship between the resistivity
and the magnetization is more complicated than ρ ∝ M 2 . This relationship
may instead be well described by an M 2 dependence of the conductivity in a
slightly more complicated circuit [150]. In the previous chapter, only the
128

Critical Transport and Magnetization of La 0.67 Ca 0.33 MnO 3
magnetic field H dependence of the resistivity is shown to be more fully
described in terms of a magnetoconductive circuit, ρ = ρ ∞ + 1/(σ 0 + σ H 2H 2 ) for
T > T C and ρ = ρ ∞ + 1/(σ 0 + σ H |H|) for T < T C , where ρ ∞ , σ 0 , σ H 2 and σ H are
parameters. In order to determine if the magnetoconductivity, σ H 2H 2 and
σ H |H| can replaced by a common term proportional to M 2 as suggested above,
the temperature dependencies and magnitudes of σ H 2, σ H and M need to be
examined.
Alternatively, it has been found in several cases that the correlation
ρ ∝ exp[-M(H,T)/M E ] is a good fit to the data [24, 114, 115]. This formulation
does not yet have a good theoretical understanding.
In this chapter, which will be published separately, the temperature and
field dependence of the magnetization and magnetoresistance, above and
below T C are reported. A well annealed La 0.67 Ca 0.33 MnO 3 thin film on a
LaAlO 3 substrate grown by MOCVD was used for transport measurements as
described in chapter 4. DC conductivity measurements were performed using
the method of Van der Pauw [52, 53]. Only the longitudinal
magnetoresistance data are shown. The transverse magnetoresistance is
nearly identical to the longitudinal as was discussed in chapter 6. After
stabilizing the temperature, repeated resistance R vs. H curves were measured
and fit with three parameters (ρ ∞ , σ 0 , and σ H 2 or σ H ) to ρ = ρ ∞ + 1/(σ 0 + σ H 2H 2 )
for T > T C and ρ = ρ ∞ + 1/(σ 0 + σ H |H|) for T < T C . The data fit well to these
forms except the few degrees near T C (where a combination of the two forms
is better) and at low temperatures. For T < 50 K the magnetoresistance shows
2
no sign of saturating (ρ = [ρ∞ + 1/σ0 ] - σH |H|/σ0 ), so only two parameters
2
([ρ∞ + 1/σ0 ] and σH /σ0 ) can be extracted.
129

130 Chapter 7
2 )
M 2 (µ B
The focus of this study is on the isotropic properties. The effects due to the
anisotropic magnetoresistance (section 6.1) and magnetocrystalline anisotropy
are minimized by concentrating on the H > 1000 Oe data for T < T C . Even in
this preliminary investigation, peculiar magnetic properties are found that
require a more complex physical model than that used for a typical
ferromagnet.
3.0
2.5
2.0
1.5
1.0
0.5
La 0.67 Ca 0.33 MnO 3 Crystal Arrott Plot
254K
258K
260K
261K
262K = Tc
263K
264K
265K
266K
268K
0.0
0 5000 10000 15000 20000
H/M (Oe/µ )
B
7.1 Magnetism near T C
Magnetization measurements were performed while the magnetic field H
was decreasing. Data for increasing H field are identical to the H decreasing
data for T > T C . Polycrystalline pellets were prepared (section 2.1.1) and this
material was used to for floating-zone laser-heated crystal growth described i n
M (µ B /Mn)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0 5 10
Magnetic Field (kOe)
15
270K
La 0.67 Ca 0.33 MnO 3 Pellet 274K
M = χH + χ 3 H 3
χ 3 > 0
272K
M = χH
274K 276K
Figure 7-1. M 2 vs. H/M plot for La 0.67 Ca 0.33 MnO 3 float zone
crystal. A mean field ferromagnet has linear isotherms with a
positive slope. The negative slope for T >T C indicates a faster
than linear increase in M (inset) due to a highly unusual
positive non-linear susceptibility χ 3 .

Critical Transport and Magnetization of La 0.67 Ca 0.33 MnO 3
2 )
M 2 × |T/T C - 1| -2β (µ B
section 2.1.2. The float zone product (referred to below as “crystal”) is dense
and crystallographically highly-oriented. The polycrystalline pellets and a
float zone crystal used for magnetization measurements have very similar
magnetic properties. A demagnetization factor N (H d = 4πNM) of about 0.5
was estimated and used to correct the data. The demagnetization correction is
large only for H small. The conclusions of this work are unaffected by
variations in N.
40
35
30
25
20
15
10
5
La 0.67 Ca 0.33 MnO 3 Scaled Arrott Plot
β = 0.27
γ = 0.90
T C =262.2K
T < T C
T > T C
0
0 100 200 300 400 500 600 700 800
H/M × |T/T - 1|
C -γ (kOe/µ )
B
Figure 7-2. Data from Figure 7-1 (using the same symbols)
scaled with β = 0.27 and γ = 0.90. According to the scaling
hypothesis, all the T < T C data should lie on a single curve
while the T > T C data should lie on a separate, single curve.
Most of the M(H, T) data can be scaled [90] with β = 0.27 and γ = 0.90.
According to the scaling hypothesis, the magnetic equation of state in the
critical region depends only on the scaled variables H/|T C /T - 1| β+γ and
M/|T C /T - 1| β . A plot of the scaled M 2 and scaled H/M, shown in Figure 7-2,
will then have only two curves: one branch for the T < T C data and another
131

132 Chapter 7
for T > T C . The regions of negative slope on the M 2 vs. H/M plot (χ 3 > 0)
make the scaled T > T C curve also approach M 2 = 0 with a negative slope. For
T - T C < 0.01 T C , the scaled data break away from this region of negative slope
to giving a nearly straight curve typical of a ferromagnet. This crossover in
scaling suggests that the true critical regime only begins with |T - T C | < 0.01
T C , and that the exponents reported here, which will be quite useful in
modeling the magnetization, are effective rather than true critical exponents.
The true critical regime is reached when the magnetic critical fluctuations
become as large as the net magnetization itself.
The experimental value of the critical exponent δ, H = M δ when T = T C ,
depends strongly on the demagnetization correction, and therefore not
analyzed here. The scaling relation δ = 1 + γ/β can be used to estimate δ. The
other critical exponents γ and β do not vary significantly when a different
demagnetization correction is used.
7. 1. 1 Spontaneous magnetization exponent
The square of the magnetization M 2 , plotted vs. H/M (Figure 7-1)
facilitates understanding the critical behavior of La 0.67 Ca 0.33 MnO 3 . The
isotherm which extrapolates to M 2 = 0, H/M = 0 is the critical isotherm T = T C .
In this way T C = 262.2 K ± 0.5 K is estimated. The uncertainty in T C is due to
the uncertainty of the demagnetization correction. The isotherms below T C
should be approximately linear and intersect H/M = 0 at M 0 . From these
M 0 (T) the (magnetic order parameter) critical exponent β ≈ 0.30 and
T C = 262.2 K can be estimated by fitting M 0 (T) ∝ (1 - T/T C ) β (Figure 7-3).
The fit is not as good as this method, in the same apparatus, allows
(Appendix A), perhaps due to the contributions in low fields from the
magnetocrystalline anisotropy, which have been ignored. Therefore, the

Critical Transport and Magnetization of La 0.67 Ca 0.33 MnO 3
values for β reported here are more uncertain than those reported for other
materials.
M 0 (µ B /Mn)
2.5
2.0
1.5
1.0
0.5
ln(M)
1.0
0.5
0.0
-0.5
M ∝ (Tc-T) β
β = 0.30
-1.0
-2 -1 0 1 2 3 4 5
ln(262.2-T)
0.0
190 200 210 220 230 240 250 260 270
Temperature (K)
Figure 7-3. Saturation Magnetization, M 0 as a function of
temperature for La 0.67 Ca 0.33 MnO 3 crystal. At each temperature,
the value shown is M extrapolated to H = 0 as given by the
intercept in Figure 7-1.Solid line is fit to M 0 (T) ∝ (1 - T/T C ) β
with β = 0.30.
Typical experimental values for the critical exponent β in Fe, Ni and YIG
[92] are 0.37 ± 0.02, which are near the theoretical values (Ising β = 0.33,
Heisenberg β = 0.36). The related metallic ferromagnets SrRuO 3 (Appendix A)
and La 0.5 Sr 0.5 CoO 3 [92] have β ≈ 0.36 and β = 0.361 respectively. Previous
magnetic data [166, 167] on La 0.67 Ca 0.33 MnO 3 have been fit with such values of
β, suggesting that the critical region has been reached. For example, muon-
133

134 Chapter 7
spin-relaxation measurements [167] give β = 0.345 ± 0.015 with T C = 274 K.
The double layer perovskite La 1.4 Sr 1.6 Mn 2 O 7 has a small [168] β = 0.25 possibly
due to 2-D ferromagnetism. The significance of the low values for β given
here for La 0.67 Ca 0.33 MnO 3 is unknown due to the uncertainty of the
measurement.
7. 1. 2 Susceptibility exponent
The T > TC isotherms should intersect M 2 = 0 at 1/χ(T, H = 0) = 1/χ0 ∝
(T/T C - 1) γ and can be used to give an effective γ ≈ 0.7 and T C = 263 K ± 1 K.
This value of γ is unexpected particularly since it is less than one (Figure 7-4).
For a typical ferromagnet, the plot of 1/χ vs. T is close to linear at high
temperatures and then become slightly concave up as T approaches T C . In
this way, the high temperature, linear extrapolation of 1/χ to 1/χ = 0 gives the
paramagnetic Curie temperature, Θ P ≈ 257 K. The ferromagnetic Curie
temperature, T C is the temperature where 1/χ and M 0 vanish, and normally
T C < Θ P . However, for γ < 1, the plot of 1/χ vs. T is concave down and Θ P < T C .
When critical fluctuations are included in the theory, γ increases due to
the suppression of the transition (Ising γ = 1.24, Heisenberg γ = 1.39). Typical
experimental values for γ (Fe, Ni and YIG [92] γ = 1.2 ± 0.2; SrRuO 3 (Appendix
A) γ = 1.17) are greater than one.
An inhomogeneous ferromagnet, with a range of T C ‘s, can provide such a
negative curvature resulting in an apparent γ < 1; however, this will also
make M 0 decrease more linearly (β ≈ 1) as T approaches T C from below. Since
the observed β is not larger but smaller than expected, a range of T C ‘s is
unlikely. In ferrimagnets such a negative curvature is expected due to the

Critical Transport and Magnetization of La 0.67 Ca 0.33 MnO 3
competition between ferromagnetic and antiferromagnetic interactions. As
pointed out by DeGennes, there is competition between double exchange
(ferromagnetic) and superexchange (antiferromagnetic) as will be discussed
further in section 7.1.4. One of the early theories of double exchange
presented by Anderson and Hasagawa [7] also predicts a downward curvature
in 1/χ vs. T.
It is not too surprising that two different values of γ (0.7 and 0.9) are
obtained from the same data. The two methods weight the data differently
and as discussed above, the true critical region where γ is a constant has
probably not been reached.
7. 1. 3 Positive nonlinear susceptibility
A peculiarity of the T > T C isotherms of Figure 7-1, which likely affects the
determination of γ, is their negative slope as they approach M 2 = 0. This slope
is proportional to the nonlinear susceptibility. For M = χH + χ 3 H 3 , χ is the
linear susceptibility and χ 3 is the third-order nonlinear magnetic
susceptibility. The slope of a T > T C isotherm in an M 2 vs. H/M Arrott plot as
M 2 approaches zero is given by -χ 4 /χ 3 . For a normal ferromagnet, χ > 0 while
χ 3 < 0 giving a positive slope for all T > T C isotherms. However in
La 0.67 Ca 0.33 MnO 3 a negative slope is found which implies that χ 3 > 0. This can
also be seen in a plot of M vs. H, which is shown in Figure 7-1. A positive χ 3
produces the faster than linear increase in M with H which leads to an
inflection point in the M vs. H curve due to the saturation of M for large H.
This effect is not likely due to the sample preparation since it is seen in both
the polycrystal and float-zone crystal samples. Arrott plots of SrRuO 3
(Appendix A) at similar temperatures do not show χ 3 > 0, which indicates that
the effect is not due to the measurement system. A more pronounced
135

136 Chapter 7
1/χ (10kOe/µ B )
5
4
3
2
1
0
260 270 280 290 300
inflection point has been observed in the double layer perovskite manganite
La 1.4 Sr 1.6 Mn 2 O 7 above T C [169].
1/χ ∝ (T-T C ) γ
γ = 0.7
Temperature (K)
A positive χ 3 can be observed in a ferromagnet or spin glass during
irreversible magnetization processes; for example, while increasing the
magnetic field from a demagnetized state. However, once the material is
magnetized at a high field, the curve of M vs. H for decreasing field almost
always has a negative χ 3 . The data in the inset of Figure 7-1 includes data for
both increasing and decreasing fields, showing no signs of hysteresis [170].
ln(1/χ)
11
10
9
1 2 3
ln(263K-T)
Figure 7-4. Inverse magnetic susceptibility, 1/χ 0 as a function
of temperature for La 0.67 Ca 0.33 MnO 3 crystal. At each
temperature, the value shown is H/M extrapolated to H = 0 as
given by the intercept in Figure 7-1. Solid line is fit to 1/χ 0 ∝
(T/T C - 1) γ with γ = 0.7 and T C = 263K.

Critical Transport and Magnetization of La 0.67 Ca 0.33 MnO 3
Thus a positive χ 3 due to the irreversibility of a spin glass seems highly
unlikely. More plausible explanations will be discussed in section 7.1.4.
7. 1. 4 Additional magnetic interaction
The unusual γ < 1 may be related to χ3 > 0. Both effects suggest an
additional mechanism. There could, for instance, be additional magnetic
interactions, such as an antiferromagnetic component to the coupling. A
canted-antiferromagnet ground-state is expected theoretically from the
double-exchange mechanism in conjunction with super-exchange [8], and has
been found by neutron diffraction [139, 143] in the related compound
Pr 0.67 Ca 0.33 MnO 3 . If there is a competition between antiferromagnetic and
ferromagnetic fluctuations in the paramagnetic state, the application of a
magnetic field will favor ferromagnetism and thus induce a larger magnetic
moment. This would explain the observed positive χ 3 . The corresponding
transition in the ordered state is called a metamagnetic transition. Competing
magnetic interactions can also cause the downward curvature in 1/χ vs. T
seen in Figure 7-4. This is commonly observed in a ferrimagnet (e.g.
Gd 0.67 Ca 0.33 MnO 3 described in section 5.1.2) where some strong interactions are
antiferromagnetic, but near T C a ferromagnetic susceptibility is observed. The
canting of the moments by as much as 45° would still give 92% of the
expected net moment, which is about that observed (section 4.1) at low
temperatures for La 0.67 Ca 0.33 MnO 3 .
A different but more general explanation for the unusual magnetic
behavior observed in La 0.67 Ca 0.33 MnO 3 requires some additional mechanism
which provides the stronger than mean field behavior, i.e. the ferromagnetic
interaction strength appears to increase, beyond that predicted by the mean
field approximation, as the magnetization increases. For instance, as T
approaches T C , the strength of the interaction increases, effectively increasing
T C . This causes the susceptibility to diverge faster than expected, providing
137

138 Chapter 7
γ < 1. Also, at a constant T > T C , as H is applied M increases, which induces
stronger coupling (higher effective T C ) and therefore increasing the
susceptibility in a nonlinear way: χ 3 > 0.
This additional mechanism may be related to the ferromagnetic metal to
paramagnetic insulator phase transition. La 0.67 Ca 0.33 MnO 3 , like other
materials which exhibit a metal-insulator phase transition, undergoes a static
and even dynamic structural transition [151, 154, 171, 172]. The changes in the
electronic and atomic structure should alter the magnetic coupling, i.e. the
exchange J is not a constant, but a function of M or ρ. Specifically, since the
metallic state has a lower volume than the insulating state, one might expect
the magnetic coupling to strengthen as the material becomes more metallic.
The metallic character, conversely, is clearly associated with the
magnetism as is predicted in the double exchange model. As the applied
magnetic field (and hence the magnetization) increases, the material becomes
more metallic. Not only does the resistivity decrease, but the metal-insulator
transition (defined by a change in sign of dρ/dT) is pushed to higher
temperatures. If the induced metallic state promotes a further increase in the
magnetization, a positive nonlinear susceptibility (χ 3 > 0) and a rapidly
diverging susceptibility (γ < 1) may be generated. Thus, the structural change
may cause the magnetism to increase faster than that expected from the mean
field model.
La 0.67 Ca 0.33 MnO 3 may have a smaller critical region than a normal
ferromagnet because of these additional interactions. In typical ferromagnets,
short range exchange interactions result in a short coherence length of the
magnetic fluctuations. These fluctuations have low energy and can be
observed relatively far from T C . In La 0.67 Ca 0.33 MnO 3 additional electronic and
lattice interactions may increase the fluctuation coherence length. With

Critical Transport and Magnetization of La 0.67 Ca 0.33 MnO 3
Magnetization (µ B /Mn)
3.0
2.5
2.0
1.5
1.0
0.5
more volume involved for each fluctuation, they will have higher energy
and only observable very near T C .
La 0.33 Ca 0.67 MnO 3 Magnetization
0.9 T C
1.1 T C
0.0
0 10 20 30 40 50 60
Internal Field (kOe)
Figure 7-5. Magnetization in a magnetic field for a
La 0.67 Ca 0.33 MnO 3 polycrystalline pellet at 0.9 and 1.1 T C . The
solid lines indicate the linear regions in each case.
7.2 Magnetoresistance
To illustrate the need for the magnetoconductivity expression described i n
section 6.2.1.1 for the magnetoresistance, the two temperatures 0.9 T C and
1.1T C are considered first. The magnetization of a polycrystalline pellet at
these two temperatures is shown in Figure 7-5. At T = 0.9 T C , M ≈ M 0 + χH
except in fields less than a few kOe when the material is not yet magnetically
saturated. At T = 1.1 T C , M ≈ χH is a good approximation particularly for
H < 40 kOe. The films used in this study have been shown to have similar
magnetic behavior as the bulk samples (chapter 4), but due to their small
volume and the complication of the substrate provide less accurate magnetic
139

140 Chapter 7
∆R/R (%)
10
0
-10
-20
-30
-40
-50
-60
ρ ∞
∆R/R
1/σ H |H|
1/σ 0
0.9 T C La 0.33 Ca 0.67 MnO 3
-M 2
-60 -40 -20 0 20 40 60
Internal Field (kOe)
data. Therefore, in what follows, the magnetotransport on films is compared
with magnetization measured on bulk (ceramic and crystal) samples.
The magnetoresistance, ∆R/R, and M 2 as functions of internal field
(making the demagnetization correction described above) are compared below
and above T C in Figure 7-6 and Figure 7-7. They are clearly related
qualitatively, although there are marked differences at high fields as
saturation is approached, particularly for T < T C .
The saturation of the magnetoresistance is best described by the fit to a
magnetoconductivity circuit (section 6.2.1.1), as if the magnetic field were
opening channels of conductivity. This circuit assumes a linear M vs. H
∆R/R (%)
-50
-55
σ = σ 0 + σ H |H|
σ = A e |H|/B
-60
40 50 60 70
Magnetic Field (kOe)
-4.5
-5.0
-5.5
-6.0
-6.5
-7.0
Figure 7-6. Magnetoresistance of La 0.67 Ca 0.33 MnO 3 film
compared with -M 2 of a pellet, both at 0.9 T C . The solid line for
the magnetoresistance data shows the fit using the indicated
equivalent circuit. The dashed line in the inset compares the
exponential fit.
-M 2 (µ B
2 /Mn)

Critical Transport and Magnetization of La 0.67 Ca 0.33 MnO 3
(section 6.2.1.1). The resistivity is fit to ρ = ρ ∞ + 1/σ where σ = σ 0 + σ H 2H 2 for
T > T C , while for T < T C , σ = σ 0 + σ H |H| is used. Each fit has three free
parameters ρ ∞ , σ 0 , and σ H 2 or σ H . If the magnetoconductivity is indeed
proportional to M 2 as suggested above, then the nonlinearity of M vs. H
(Figure 7-5) will slightly alter the fitting parameters, particularly ρ ∞ .
∆R/R (%)
0
-8
-16
-24
-32
-40
-48
1.1 T C La 0.33 Ca 0.67 MnO 3
∆R/R -M 2
ρ ∞
1/σ H H 2
2
1/σ 0
-60 -40 -20 0 20 40 60
Internal Field (kOe)
Figure 7-7. Magnetoresistance of La 0.67 Ca 0.33 MnO 3 film
compared with -M 2 of a pellet, both at 1.1 T C . The solid line for
the magnetoresistance data shows the fit using the indicated
equivalent circuit.
The exponential dependence of the resistivity first proposed by Hundley et
al. [114] on the magnetization ρ = ρ E exp[-M(H,T)/M E ] does not fit the
magnetoresistance data as well above T C as the fit described above. For T > T C
and H small, ρ ≈ ρ E exp[-χH/M E ] ≈ ρ E (1 - χH/M E ) predicts a magnetoresistance
linear in H while the observed magnetoresistance is quadratic (Figure 7-7).
Below T C the exponential fit is not at good as the magnetoconductivity circuit
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-M 2 (µ B
2 /Mn)
141

142 Chapter 7
(Figure 7-6), in so far as the H dependence is concerned; however, the
exponential dependence upon M is found, as will be shown below.
ρ ∞ (mΩcm)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
ρ ∞
σ 0 and ρ ∞ above T C
280 290 300 310 320 330
Temperature (K)
Figure 7-8. Fitting parameters σ 0 and ρ ∞ for T > T C in a
La 0.67 Ca 0.33 MnO 3 film. The temperature dependence of these
two parameters reflect the insulating behavior of the material.
7. 2. 1 Magnetoresistance scaling above T C
The magnetoresistance data at various temperatures above T C were fit to
the expression ρ = ρ ∞ + 1/(σ 0 + σ H 2H 2 ). The resulting parameters σ 0 and ρ ∞ are
shown in Figure 7-8. The parameter σ 0 remains relatively constant, slightly
increasing as expected for a semiconductor. The significance of ρ ∞ will be
discussed below.
σ 0
0.20
0.15
0.10
0.05
0.00
σ 0 (mΩcm -1 )

σ (10
H -10 mΩcm -1 Oe -2 2
)
Critical Transport and Magnetization of La 0.67 Ca 0.33 MnO 3
2.0
1.5
1.0
0.5
0.0
ln(σ 2)
Η
-22
-23
-24
-25
-26
280 300 320 340 360
Temperature (K)
The parameter σ H 2, which determines the magnetoresistance, is highly
temperature dependent (Figure 7-9), diverging at T C . The data can be fit with
a critical exponent of -2.0 ± 0.1: σ H 2(T) = σ H 2(2T C ) (T/T C - 1) -2 , where T C = 262 K ±
1 K and σ H 2(2T C ) ≈ 8 × 10 -13 Oe -2 mΩcm -1 . This result supports an M 2
dependence of the magnetoconductance (or magnetoresistance) since
M 2 ∝ (T/T C - 1) -2γ above T C , where, as experimentally determined above,
γ ≈ 0.9. Specifically, it is found that σ(H,T) = σ 0 + σ H 2(T)H 2 ≈ σ 0 + σ M 2M 2 . Using
the Curie Weiss susceptibility from [103], M = χH ≈ 5 × 10 -4 (emu G -1 cm -3 ) ×
(T/T C - 1) -1 H, gives σ M 2 = 3 × 10 -6 Oe -2 mΩcm -1 .
Slope = -2
σ ∝ χ
Η 2
2
Above T
C
Resistance ≈ 1/(1+M 2 )
2.8 3.2 3.6 4.0 4.4
ln(T-262K)
Figure 7-9. Fitting parameter σ H 2 as a function of temperature
in a La 0.67 Ca 0.33 MnO 3 film for T > T C . The temperature
dependence of σ H 2 and the square of the susceptibility are the
same, indicating a relationship between the
magnetoconductance and M 2 .
143

144 Chapter 7
The related form ρ ∝ -M 2 would also be consistent with our low field data
since in low fields ∆ρ = -H 2 2
σH 2/σ0 and σ0 is nearly constant with respect to
temperature. This form was also observed above T C in La 1-x Sr x CoO 3 [165] and
in small fields for lower T C films [158] of La 0.7 Ca 0.3 MnO 3 . In the temperature
range of this experiment, T > 1.01 T C , 4πM < H, so B = H + 4πM ≈ H and
therefore not very temperature dependent. Thus the resistivity or
conductivity is not proportional to B or B 2 .
7. 2. 2 Magnetoresistance scaling below T C
Below T C , ρ ∞ can be interpreted as due to scattering processes which would
persist if the moments were perfectly ordered. As shown in Figure 7-10, the
resistivity in a infinite field, ρ ∞ , follows the A + BT 2 fit to the low-
temperature, H = 0, intrinsic resistivity described in section 4.2.1. In a true
infinite field (H = ∞) one would expect the resistance to be continuous with a
positive (metallic) slope as T increases through T C . While there is a factor of 2
discrepancy between ρ ∞ (T C ) extrapolated from above (Figure 7-8) and below
(Figure 7-10) this is not unreasonable because two different fitting equations
are used. Particularly above T C , the fitting parameter ρ ∞ may not reflect the
true H = ∞ resistivity because the nonlinearity of M upon H is not taken into
account.
The parameter σ 0 , which can be related to the resistivity in zero field
ρ(H = 0) ≈ ρ ∞ + 1/σ 0 , describes the limiting field-dependent, spin-disorder
scattering processes present as H approaches 0. Unlike ρ ∞ , 1/σ 0 diverges as the
Curie temperature is approached. A critical exponent of about 1.8 (Figure 7-
10) is found: σ 0 (T) ≈ σ 0 (0) (1 - T/T C ) 1.8 , where σ 0 (0) ≈ 2 × 10 -3 mΩcm -1 .

ρ ∞ and 1/σ 0 (mΩ cm)
Critical Transport and Magnetization of La 0.67 Ca 0.33 MnO 3
2.5
2.0
1.5
1.0
0.5
ln(σ 0 )
4
3
2
1
0
-1
σ 0 ∝ (T C -T) 1.8
-2
2 3 4 5
ln(262K-T)
σ 0 and ρ ∞ below T C
0.0
80 100 120 140 160 180 200 220 240
Temperature (K)
Figure 7-10. Fitting parameters σ 0 and ρ ∞ for T < T C in a
La 0.67 Ca 0.33 MnO 3 film. ρ ∞ is governed by the A + BT 2 terms in
the resistivity while σ 0 diverges at T C . The inset shows σ 0 data
fit with a (T C - T) 1.8 power law (dashed line), and σ ∝
exp(M/M E ) (solid line). The zero field resistivity ρ(H = 0) =
ρ ∞ + 1/σ 0 is shown for comparison.
The parameter σ H is more difficult to interpret than the other parameters.
2
Although the magnetoresistance, ∆R/HR proportional to σH /σ0 , increases as
T C is approached, σ H decreases. Thus the large magnetoresistance found
below T C is due to the divergence of 1/σ 0 and not σ H . The parameter σ H does
not always appear to vanish completely as T approaches T C , making it difficult
to fit to a (1 - T/T C ) n power law. Assuming T C = 262K (determined from the
critical properties of σ H 2 and σ 0 ) the exponent for σ H is about 0.7.
ρ ∞
1/σ 0
ρ(H = 0)
A + BT 2 fit
145

146 Chapter 7
σ H (10 -5 mΩcm -1 Oe -1 )
20
15
10
5
0
ln( σ H )
100 150 200 250
Temperature (K)
Figure 7-11. Fitting parameter σ H as a function of temperature
in a La 0.67 Ca 0.33 MnO 3 film for T < T C . The solid line shows the
best fit to the data using a critical exponent of 0.7.
There is no simple relationship between either the magnetoconductance
or the magnetoresistance with M 2 below T C because the measured scaling
exponents do not agree with those predicted from an M 2 dependence as
discussed in the following. Shown in Figure 7-5 is the magnetization below
T C which can be approximated with M(H) = M 0 + χH , and therefore
M 2 2
≈ M 0 + 2χM0H. Assuming σ(H,T) = σ0 + σHH ≈ σM 2 M(H,T) 2 as suggested
2
above, then below TC the following relations should hold: σ0 = σM 2M0 and
σ H = 2χσ M 2M 0 . The temperature dependence of M 0 and χ below T C should be
governed by the scaling exponents β and γ, via M 0 ∝ (1 - T/T C ) β and
χ ∝ (1 - T/T C ) -γ where experimentally β = 0.3 and γ ≈ 0.9 (typical critical
exponents are β ≈ 0.35 and γ ≈ 1.2). Thus one would expect the critical
-9
-10
-11
3.0 4.0 5.0
ln(262K-T)

Critical Transport and Magnetization of La 0.67 Ca 0.33 MnO 3
exponent for σ 0 and σ H to be 0.6 and -0.6 respectively. Experimentally,
however σ 0 ∝ (1 - T/T C ) 1.8 and σ H ∝ (1 - T/T C ) 0.7 .
An M 2 dependence of the magnetoresistance (as opposed to the
magnetoconductance) where ρ ≈ ρ0 - ρMM 2 2
≈ (ρ0 - ρMM0 ) - 2ρMχM 0H below TC ,
2
is also inconsistent. For low fields ρ = (ρ∞ + 1/σ0 ) - (σH /σ0 )|H| is found
2
experimentally, which has a field dependent term (σH /σ0 ), which varies as
(1 - T/T C ) -2.9 . This is quite different from 2ρ M χM 0 which has a critical exponent
of β - γ ≈ -0.6.
Below T C , in agreement with previous work [23, 107], the conductivity is
exponentially dependent upon M. Removing the slowly varying
contribution due to ρ ∞ , this model predicts σ(H,T) = σ E exp[M(H,T)/M E ] ≈
σ 0 + σ H H. For H = 0 this reduces to σ 0 = σ E exp[M 0 (T)/M E ]. The experimental
relationship M 0 (T) ∝ (1 - T/T C ) β with β = 0.3 can be used to fit the magneto-
resistance data. The higher quality of this fit compared to the M 2 fit is shown
in Figure 7-10, with M E ≈ 0.4 µ B (M E = 1.0 µ B in [107]) and σ E ≈ 4 × 10 -3 mΩcm -1 .
An exponential dependence of the conductivity may suggest a tunneling
mechanism is responsible. Tunneling conductivity depends exponentially on
the length of the tunneling barrier. If, in some way, this barrier is decreased
by an increase in the magnetization, then the conductivity will depend
exponentially on M as observed for large M. Spin dependent tunneling is
reported to be the mechanism of the large domain boundary
magnetoresistance observed in these materials [140].
Furthermore, the temperature dependence of σ H can also be explained
with the exponential model. According to this model, the field dependent
conductivity
d d M M
σH σ σE
dH dH
σσ E σE σχ
e
/ E
= = 0 dM
= = 0
M dH M
(where
E
E
147

148 Chapter 7
σ E exp[M 0 (T)/M E ] ≈ σ 0 is used) should have the same temperature dependence
as σ 0 χ. Experimentally, σ 0 χ has an approximate temperature dependence of
(1 - T/T C ) 1.8-0.9 , assuming γ ′ (susceptibility exponent for T < T C ) = γ. This is
within experimental error of the critical exponent found (Figure 7-11) for σ H
(about 0.7).
7. 2. 3 Magnetoresistance scaling at T C
At or very near T C , the magnetic properties should be described by the
critical exponent δ, where M ∝ H 1/δ when T = T C . The magnetoresistance data
for T = 262 K ≈ T C is shown in Figure 7-12, and can be fit to ρ = ρ ∞ +
1/(σ 0 + σ H nH n ), where n is an additional free parameter. The data fit well with
n = 1.2 ± 0.1, where the variation of n arises from different ranges and
weighting of the fit. Thus, if at T C , the conductivity σ ∝ M 2 then σ ∝ H 2/δ , or
δ = 2/n. This is clearly not the case since 2/n ≈ 1.6 while typically δ ≈ 4.8, or
from the data presented above and the scaling relation δ = 1 + γ/β the value
δ ≈ 4.3 is found.
The T = 262 K data fit well to the composite relation proposed by Sun et al.
[23] σ ∝ M 2 exp[M/M E ] combined with H = M δ . Above T C this relation reduces
to σ ∝ M 2 , which, as was shown above, is quite accurate. Below T C , the
exponential term dominates invalidating the simple σ ∝ M 2 relationship.

Critical Transport and Magnetization of La 0.67 Ca 0.33 MnO 3
∆R/R (%)
0
-10
-20
-30
-40
-50
-60
-70
-80
ρ ∞
262 K ≈ T C La 0.33 Ca 0.67 MnO 3
n = 1.3
1/σ H H n
n
1/σ 0
-60 -40 -20 0
Field (kOe)
20 40 60
Figure 7-12. Magnetoresistance of La 0.67 Ca 0.33 MnO 3 film at 262
K ≈ T C . The solid line shows the fit (for the full data on a
linear scale) using the indicated equivalent circuit.
7. 2. 4 Relation to low temperature magnetoresistance
This qualitative correlation between the magnetization and the resistivity
may also account for the intrinsic, small, linear magnetoresistance (Figure 4-
8) found even at the lowest temperatures where the magnetization is nearly
saturated. If at these low temperatures σ = σ 0 + σ M 2M 2 then ∆σ/σ (or ∆ρ/ρ) is
given by ∆σ = 2χσ M 2M 0 H. Since ∆σ = -∆ρ/ρ 2 , the susceptibility χ required to
give the observed magnetoresistance is χ = -∆ρ/(2ρ 2 σ M 2M 0 H ) = 2.5 × 10 -4
emu/Oe cm 3 using the results from chapter 4 for ∆ρ/H = -1.5 × 10 -8 mΩcm/Oe,
ρ = 0.125 mΩcm, and M 0 = 3.4 µ B /Mn = 550 emu/cm 3 . This is within a factor
of three of the observed value (section 4.2.9, Figure 4-8) χ = 9 × 10 -5
emu/Oe cm 3 . Similarly acceptable, is the exponential model
149

150 Chapter 7
σ = σ E exp[M(H,T)/M E ], which gives at low T and H,
∆ρ/H = -χexp[-M 0 /M E ]/M E σ E ; requiring χ = 32.9 × 10 -5 emu/Oe cm 3 . A mag-
2 2 2
netoresistive model ρ = (ρ∞ + 1/σ0 ) - (σM 2/σ0 )M giving ∆ρ = -2χσM2M0H /σ0 is
2
significantly worse since it would predict χ = -∆ρσ0 /(2σM2M0H ) = 2 × 10 -7
emu/Oe cm 3 .
In a similar manner, the correlation between the magnetization and the
resistivity should contribute to the T 2 contribution (section 4.2.1.2) to the low
temperature resistivity ρ = R 0 + R 2 T 2 + … . This is because the magnetization
can be approximated with M = M 0 (1 - (T/Θ) 2 ) for these temperatures (where
Θ ≈ 500 K, section 4.1.1), giving ρ(T) ∝ -M(T) ∝ T 2 . For the σ = σ 0 + σ M 2M 2
model ρ(T) - ρ(0) = -ρ 2 (σ(T) - σ(0)) = 2σM 2ρ 2 2 2 -7 -2 2
M0 (T/Θ) ≈ 1.26 × 10 mΩcm K T .
For the ρ = exp[-M(H,T)/M E ]/σ E model ρ(T) - ρ(0) = exp[-M 0 /M E ]/σ E ×
(T/Θ) 2 M 0 /M E ≈ 1.14 × 10 -6 mΩcm K -2 T 2 . Both of these proposed contributions
are less than the observed value (section 4.2.1.2) R 2 ≈ 1.96 × 10 -5 mΩcm K -2 ,
suggesting that scattering mechanisms such as electron-electron, or electron-
magnon scattering dominates the R 2 T 2 term.
7.3 Conclusion
A scaling relation has been found that satisfactorily characterizes the
magnetic and magnetoresistive properties of La 0.67 Ca 0.33 MnO 3 near T C .
However the scaling exponents, particularly γ < 1, show that critical scaling
has not been reached even though the measurements reach 0.01 T/T C where
simple ferromagnets are expected to exhibit critical scaling.
An unusual positive nonlinear magnetic susceptibility, which does not
disrupt the scaling, is also observed. The positive non-linear susceptibility
and susceptibility exponent γ < 1 may result from the coupling of the

Critical Transport and Magnetization of La 0.67 Ca 0.33 MnO 3
ferromagnetic and metal-insulator transitions, making the exchange J a
function of M.
The scaling of both the H and T dependence of the magnetoconductance
above T C is consistent with a simple M 2 dependence of the conductivity.
However, below or very near T C this correlation predicts the observed H
dependence but can not account for the T dependence. To fit these data, the
model for the magnetoresistance needs to have an exponential dependence
on the magnetization for large M, which reduces to M 2 for small M , such as
that previously suggested by Sun et al. [23] σ ∝ M 2 exp[M/M E ].
151

Appendix A. Critical Behavior and Anisotropy in Single
Crystal SrRuO 3
Introduction
Strontium ruthenate, SrRuO 3 , has many physical properties which make
it unique among perovskite oxides. First, it is metallic in the undoped state
[36, 126, 173-176]. This is even more unusual since the ruthenium in SrRuO 3
is in a high oxidation state; whereas many other metallic perovskites must be
formed in reducing environments and therefore unstable in air at high
temperatures. The remarkable chemical stability and simple chemical
formula makes metallic SrRuO 3 quite attractive for use in epitaxial thin film
heterostructures with other perovskite oxides when metallic layers are
desired. Indeed, the nearly cubic SrRuO 3 [177] is often preferred over the
more distorted CaRuO 3 when making structures such as electrodes for
ferroelectrics or superconductor-normal metal-superconductor junctions.
The other striking feature of SrRuO 3 is its ferromagnetism with a
reasonably high transition temperature (163 K) and large saturation moment
(> 1µ B ) [174, 175, 178, 179]. SrRuO 3 is the only ferromagnetic perovskite oxide
of a 4d or 5d transition metal. Moreover, SrRuO 3 has the largest saturation
moment known to arise from 4d electrons, making it more related to the iron
group ferromagnetic metals (Fe, Co and Ni) than to the weak itinerant
electron ferromagnets such as ZrZn 2 . SrRuO 3 also has very strong cubic
magnetic anisotropy, requiring magnetic fields in excess of 10 Tesla to saturate
the magnetization in the hard directions. Such a strong anisotropy makes
measuring even the simplest properties, such as saturation magnetization,
difficult. It is the purpose of the present work to determine the magnetic
properties of SrRuO 3 by measurements of magnetically-soft single crystals
along the easy magnetic direction.
152

Critical Behavior and Anisotropy in Single Crystal SrRuO 3
There have been several studies of SrRuO 3 in the past three decades,
mostly on polycrystalline samples which are quite easy to prepare. SrRuO 3
has a very slightly distorted (GdFeO 3 type) perovskite structure. The
deviation from perfect cubic perovskite is so small that it has often been
undetected. The resistivity of polycrystalline, and epitaxial thin film SrRuO 3
shows a cusp in dρ/dT at the ferromagnetic Curie temperature T C . This is
commonly observed for metallic ferromagnets and is attributed to spin
disorder scattering [125]. Reported T C 's tend to vary from 150K to 165K.
The saturation magnetization of SrRuO 3 has been both difficult to
measure and interpret. Low spin Ru 4+ in an octahedral coordination has four
4d electrons in the t 2g triply degenerate state, giving two paired and two
unpaired electrons. Since the orbital component of angular momentum J
will be quenched, J = S = 1 is expected. The measured Curie constant of the
paramagnetic state is consistent with this model (expected: µ eff = √2gJ(J + 1) µ B
= 2.83 µ B ; measured = 2.67 µ B [175]). For a localized moment ferromagnet, the
saturation magnetization M S is predicted to be M S = gJ µ B = 2.0 µ B for SrRuO 3 .
Measured values of M S are much less. Polycrystalline SrRuO 3 reaches about
0.85 µ B [175, 178, 179] in low fields but continues to increase in higher
magnetic fields. At 125 kOe it was noted that M had reached 1.55 µ B but had
not yet saturated [178]. Early neutron diffraction derived a moment of 1.4 ±
0.4 µ B [178] with no evidence for any antiferromagnetic order (spin canting).
Recent theoretical investigations predict incomplete band splitting and a large
moment of about 1.6 µ B [176, 180]. Early explanations for the low value of M S
included spin canting, band magnetism, and incomplete alignment of the
magnetization due to magnetocrystalline anisotropy [178]. The present work
shows that SrRuO 3 has a large saturation moment of 1.6 µ B as predicted by
these calculations.
Single crystals of SrRuO 3 can be grown from a SrCl 2 flux [36]. The
resistivity of such crystals is consistent with the results on polycrystalline
153

154 Appendix A
samples. Magnetization and magnetic torque measurements of single crystals
[181-183] showed that SrRuO 3 has a high cubic anisotropy with 〈110〉 (cubic
cell) being the easy direction, with nearly square hysteresis loops. Some
previous measurements reported for single crystals are questionable, for
example, the measured value of M S = 1.1 µ B . In this work new magnetization
data is shown to clarify these points.
Experimental
Single crystals of SrRuO 3 were grown by slow cooling in a SrCl 2 flux [36,
184]. Polycrystalline SrRuO 3 was prepared from stoichiometric quantities of
SrCO 3 and Ru metal repeatedly reacted at 1260°C, and was used as the source
material for crystal growth. The SrCl 2 was dried in air at 110°C. A mixture
with approximate weight ratio 1:20 of SrRuO 3 :SrCl 2 was melted in a platinum
crucible with lid at 1260°C for 94 hrs. The sample was cooled to 800°C at 1°/hr
and then to room temperature at ~40°/hr. Crystals of SrRuO 3 less than 1 m m
in diameter were found at the bottom of the crucible after removing the flux
with water. Most crystals were cubo-octahedron shaped and grew with a
3-fold symmetric axis (presumably [111]) perpendicular to the Pt surface. The
(cubic) crystal orientation was determined by the symmetry of the faces. The
actual orthorhombic symmetry [177] was confirmed by powder x-ray
diffraction and Transmission Electron Microscopy.
Magnetization in fields up to 70 kOe was measured using a Quantum
Design MPMSR 2 SQUID magnetometer. The samples were attached to a
plastic straw with a small amount of Apiezon N vacuum grease. Samples
were oriented on the straw visually using crystal faces which had obvious 2-,
3- and 4-fold rotational symmetry. At room temperature, the crystal can
rotate in a large field to align a paramagnetic easy direction with the field.
This was used as a final adjustment when aligning along the easy 〈110〉

Critical Behavior and Anisotropy in Single Crystal SrRuO 3
directions. At low temperatures, the grease solidifies so the sample cannot
rotate. The accuracy of the magnetization measurement was estimated by
measuring a sphere of yttrium iron garnet described in section 3.2.1.1. For the
measure-ments reported here, the sample holder was fixed at the rotation
angle which gave the maximum magnetization. Several crystals were
measured.
Results
Magnetic Moment (µ B /Ru)
1.7
1.6
1.5
1.4
1.3
M S
√(2/3) M S
SrRuO 3 Single Crystal
[110]
[1-10]
[101]
[111]
[100]
1.2
1.1
0
M /√2
S
10 20 30
-2
-75
40
-50 -25 0 25 50
H (kOe)
50 60
75
70
Magnetic Field (kOersted)
Figure A- 1. Magnetization at 5 K of SrRuO 3 single crystal
along several crystallographic directions showing strong cubic
but not uniaxial magnetocrystalline anisotropy. Inset shows
the full hysteresis loop of the single crystal data along with
that of a polycrystalline pellet for comparison.
Figure A- 1 shows hysteresis loops at 5 K for SrRuO 3 single crystal i n
various crystallographic directions and a polycrystalline pellet. The crystals
have a low coercive field (≈ 10 Oe) compared to that of the polycrystalline
pellet (3000 Oe). But more importantly, the crystals show very little hysteresis
M (µ B /Ru)
2
1
0
-1
Pellet
155

156 Appendix A
while the pellet displays noticeable hysteresis even in fields greater than
40 kOe.
The rapid, linear approach to saturation (with respect to the applied field)
found in all directions can be attributed to demagnetization. Until the sample
becomes fully magnetized, the demagnetization field H d is equal to the
applied field H a resulting in an internal field of zero (H i = H a - H d ). From this
slope (M = H a /4πN), one can calculate the demagnetization factor N and
therefore calculate the internal field. The measured demagnetization factors
N, (H d = 4πNM) are 0.25, 0.28, 0.31, 0.49, 0.66 for the [110], [1-10], [101], [100] and
[111] directions respectively. These seem reasonable considering the shape of
the crystals. Since a uniaxial magnetocrystalline anisotropy will also give a
linear increase in M if H is applied along the hard direction, it is difficult to
distinguish it from the effect of the demagnetization field in this study. Since
the demagnetization field can be as large as a thousand Oersted, one can only
conclude that the uniaxial anisotropy field is less than a thousand Oersted,
which is considerably less than that ( > 50 kOe) reported previously [182].
In the [110] direction SrRuO 3 rapidly approaches saturation and then
remains relatively constant (square hysteresis loop), as is expected for a
magnet with a magnetic field along the easy direction. The crystal was also
measured in the [1-10] and [101] directions which would be equivalent by
symmetry to the [110] if the crystals were cubic. Since the magnetization is the
same along these 〈110〉 type directions, it can be concluded that the magnetic
properties of single crystal SrRuO 3 are essentially cubic, i.e. only cubic
magnetocrystalline anisotropy is detected. Beyond this initial saturation
along the easy 〈110〉 directions, there is a small but measurable increase in the
magnetization which is linear in magnetic field and has a slope of 6 × 10 -7
µ B /Oe.

Critical Behavior and Anisotropy in Single Crystal SrRuO 3
2 )
M 2 (µ B
1.0
0.8
0.6
0.4
0.2
SrRuO 3 Crystal Arrott Plot
140K
145K
152K
Log(H)
156K
158K
2
160K 1
162K
-0.9
163K = Tc
-0.8 -0.7 -0.6 -0.5
Log(M)
-0.4 -0.3
164K
166K
168K
170K
0.0
0 5000 10000 15000
H/M (Oe/µ )
B
20000 25000 30000
The H = 0, T = 0 saturation moment M S found along the easy 〈110〉
directions are about 1.62 µ B /Ru. In the remnant state (H reduced to zero), the
magnetization should lie along the nearest easy direction. Thus the expected
remnant (H = 0) magnetization along the [100] or [111] direction is simply the
cosine of the angle it makes with the closest 〈110〉 direction. For 〈100〉 the
expected remnant magnetization is M S /√2, and for 〈111〉 directions √(2/3)M S is
expected. These are extremely close to the experimental values (Figure A- 1).
In the other primary directions, there is a nonlinear approach to satur-
ation which is characteristic of materials with cubic anisotropy. The magneto-
crystalline anisotropy constants K 1 and K 2 can be estimated from these
magnetization curves [185]. The magnetocrystalline anisotropy energy E can
be defined in terms of the cosine of the angle M makes with the three crystal
2 2 2 2 2 2
axes: αi = xi • M/M. For a cubic material E = K1 (α1 α2 + α2 α3 + α3 α1 ) +
4
3
T = 163K H ∝ M δ
δ = 4.2(1)
Figure A- 2. Arrott Plot of SrRuO 3 single crystal along easy
[110] direction. Inset, critical isotherm (T = 163K ≈ T C ) on a log
scale fit to M δ ∝ H with δ = 4.2.
157

158 Appendix A
M 0 (µ B )
1.0
0.8
0.6
0.4
0.2
Log(M 0 )
2 2 2
K2 (α1 α2 α3 ) + … . In SrRuO3 the magnetic easy axes are 〈110〉, which requires
K 1 < 0. Similarly, 〈100〉 are the hard axes (〈111〉 are intermediate) which
implies 2.25|K 1 | < K 2 < 9|K 1 |. The [100] magnetization should intersect that
of the easy axis [110] at H a = -2K 1 /M S . An extrapolation of the [100] M vs. H
curve gives -2K 1 /M S ≈ 109 kOe. Alternatively, the area between the [100] and
the [110] M vs. H curves should be equal to -K 1 /4. This method gives a value
for -2K 1 /M S ≈ 96 kOe. The area method can also be used with the [111] curve
to estimate 2K 2 /M S ≈ 540 kOe.
0.0
-0.1
-0.2
-0.2
-0.3
-0.4
-0.5
SrRuO 3 M 0 ∝ (T C -T) β
β = 0.32
T far from T C
β = 0.39
T near T C
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Log(T -T)
C
Average β = 0.36
T C = 163.2(1) K
0.0
125 130 135 140 145 150 155 160 165
Temperature (K)
Figure A- 3. Zero field magnetization M 0 of SrRuO 3 single
crystal along easy [110] direction. Solid line shows the fit to
M 0 (T) ∝ (1 - T/T C ) β with β = 0.36. Inset showing the same data
on a log plot. The critical exponent β appears to change from
Heisenberg-like β = 0.39 near T C to Ising-like β = 0.32 as T
decreases.
In the easy [110] direction, M vs. H curves were measured for various tem-
peratures and are shown in Figure A- 2 as M 2 vs. H/M (Arrott plot). The iso-
therms below T C should be approximately linear and intersect H/M = 0 at M 0 .

Critical Behavior and Anisotropy in Single Crystal SrRuO 3
1/χ 0 (emu -1 mol Gauss)
25
20
15
10
5
Log(1/χ 0 )
5.0
4.5
4.0
3.5
3.0
SrRuO 3 Crystal Inverse Susceptibility
- γ
χ ∝ (T-T )
0 C
γ = 1.17
0 0.5 1 1.5
Log(T-T )
C
0
150 200 250 300 350
Temperature (K)
0
160 165 170 175 180 185 190 195
Temperature (K)
From these M 0 (T), shown in Figure A- 3, the critical exponent β ≈ 0.36 and
T C = 163.2 ± 0.2 K can be estimated by fitting M 0 (T) ∝ (1 - T/T C ) β in the critical
region. The fit with a single value for β = 0.36 is poor considering the
apparent precision of the data. Closer to T C , the data fit better with a larger
β ≈ 0.39 which is close to that predicted in the 3-d Heisenberg model (β = 0.38).
Farther from T C , the exponent is smaller, β ≈ 0.32, near that predicted by the
Ising model (β = 0.33). It is possible that this is due to a crossover from
Heisenberg to Ising behavior at T decreases from T C . A similar model has
been proposed to explain measurements of thin film SrRuO 3 [73].
The T > T C isotherms of Figure A- 2 should intersect M 2 = 0 at
1/χ(T, H = 0) = 1/χ 0 . The critical exponent γ is then found from 1/χ 0 ∝
1/ χ 0 (emu -1 mol Gauss)
200
150
100
50
Pellet
Single Crystal
Figure A- 4. Zero field inverse susceptibility 1/χ 0 of SrRuO 3
single crystal along easy [110] direction. Solid line shows the
fit to 1/χ 0 (T) ∝ (1 - T/T C ) γ with γ = 1.17 and T C = 163.2 K. The
inset shows the same data on a log plot.
159

160 Appendix A
(T/T C - 1) γ . From these data (Figure A- 4) γ = 1.17 ± 0.02 is estimated. The plot
of 1/χ vs. T should be close to linear for high temperatures.
2 )
M 2 × |T/T C - 1| -2 β (µ B
6.0
5.0
4.0
3.0
2.0
1.0
SrRuO 3 Crystal Scaled Arrott Plot
T < T C
T > T C
0.0
0 500 1000 1500 2000 2500
H/M × |T/T - 1|
C -γ (kOe/µ )
B
The M(H, T) data can be replotted using the scaling hypothesis [90] with
β = 0.36 and γ = 1.17. According to the hypothesis, the magnetic equation of
state in the critical region depends only on the scaled variables H/|T C /T - 1| β+γ
and M/|T C /T - 1| β . A plot of the scaled M 2 and scaled H/M, shown in
Figure A- 5, will then have only two curves: one branch for the T < T C data
and another for T > T C . Not all the curves fit on a single line, this is due to
the apparent change in β as discussed above.
β = 0.36
γ = 1.17
T C =163.2K
Figure A- 5. Scaled Arrott Plot of SrRuO 3 single crystal along
easy [110] direction with β = 0.36 and γ =1.17. Symbols are the
same as those used in Figure A- 2.
The critical isotherm (T = T C ) should obey the relation M δ ∝ H, where
according to the scaling relation δ = γ/β + 1. From the previously measured
values of β and γ, the exponent δ is therefore expected to be 4.3 ± 0.5. This is
in good agreement with the measured value (Figure A- 2) δ = 4.2 ± 0.2.

Critical Behavior and Anisotropy in Single Crystal SrRuO 3
Magnetization (µ B /Ru)
The temperature dependence of M along the easy direction is shown in
Figure A- 6 for various applied fields. The demagnetization field of 4πNM ≈
800 Oe should be subtracted from the applied field to give the internal field.
Between 5 K and 100 K, and for internal fields from 0.2 kOe to 68 kOe the
magnetization is well approximated by M = M 0 (1 - (T/Θ 2 ) 2 ) where M 0 and Θ 2
are fitting parameters. Other possible analyses are discussed below.
Discussion
1.65
1.60
1.55
1.50
1.45
1.40
1.35
M (µ B /Ru)
SrRuO 3 Single Crystal along easy direction [110]
1.7
1.6
1.5
1.4
1.3
0 2000 4000 6000 8000 10000
T
1.30
0 10 20 30 40 50 60 70 80 90 100
Temperature (K)
2 (K 2 )
Figure A- 6. Magnetization as a function of temperature of
SrRuO 3 single crystal along easy [110] direction. Inset shows
the approximate T 2 dependence of the magnetization.
The low temperature magnetization data (Figure A- 1) of bulk SrRuO 3 can
be explained with 〈110〉 easy directions and a large cubic, but very little
uniaxial, magnetocrystalline anisotropy. The low coercivity of the crystals
(~10 Oe), compared to the 3 kOe coercivity found in polycrystalline samples,
7T
6T
5T
4T
3T
2T
1T
2kG
1kG
161

162 Appendix A
provides highly reversible and square hysteresis curves. There is no
indication of a magnetic multi-domain structure [183].
The cubic magnetocrystalline anisotropy in SrRuO 3 is very large, as
reported previously [181, 183]. Typical values of K 1 for cubic 3-d ferromagnets
are 100 times smaller than that found for SrRuO 3 . Such large values of K 1
usually refer to uniaxial anisotropy, for instance in hexagonal materials,
which is a lower order effect (K 1 refers to the first non-zero anisotropy
constant). This large anisotropy probably results from the strong spin-orbit
coupling of the heavy Ru atom, which also gives SrRuO 3 a strong Kerr effect
[186].
From measurements along different 〈110〉 directions, no indication of
uniaxial magnetocrystalline anisotropy [182] is found, although it is allowed
from the orthorhombic symmetry. This might be expected since the
crystallographic unit cell lengths [177] vary by only 0.03%, and the angles by
0.4% from the perfect cubic ones. The distortion from cubic is primarily due
to a rotation of the RuO 6 octahedra, which alters the symmetry much more
than the shape of the unit cell [177]. In the crystals reported here, the
orthorhombic cell was confirmed using TEM. Because the unit cell is only
slightly distorted, the few reports claiming cubic [184] or tetragonal [182]
crystallographic symmetry without supporting evidence, should be
reevaluated in this context. Significant uniaxial anisotropy reported in thin
films of SrRuO 3 [187] may result from growth induced anisotropy, as was
found in films of magnetic garnets used in magnetic bubble technology [188].
Due to the large magnetocrystalline anisotropy, the saturation moment
of SrRuO 3 is difficult to measure. In directions other than 〈110〉 the magnetic
moment does not saturate even in fields of several 10 kOe. The strain and
small particle size of a polycrystalline sample apparently makes it even more
difficult to saturate than the hard direction in a single crystal (Figure A- 1).
Polycrystalline SrRuO 3 has a remnant magnetization (H = 0) of M S ≈ 0.85 µ B ,

Critical Behavior and Anisotropy in Single Crystal SrRuO 3
which increases non-linearly past 1.55 µ B (H = 125 kOe) [178]. Clearly a
magnetically soft single crystal with H along the easy direction is needed to
measure M S . In a previous experiment [181] on single crystal SrRuO 3 with a
square hysteresis loop, M S = 1.1 µ B at H = 0 was reported. However, that value
of M S is clearly too small since by 17 kOe it is smaller than the value for a
polycrystalline sample [181]. Our measured value of M S = 1.6 µ B (H = 0) is
consistent with the high field polycrystalline results.
The magnetic critical exponents measured here are in the range typically
seen in large moment ferromagnets and expected theoretically for 3-
dimensional ferromagnets. Experimental values for the critical exponent β in
Fe, Ni and YIG [92] are 0.37 ± 0.02, which are near the theoretical values (Ising
β = 0.33, Heisenberg β = 0.36). The related metallic ferromagnets La 0.5 Sr 0.5 CoO 3
[92] has β = 0.361. The apparent decrease in β as T decreases from T C , may be
due to the large magnetocrystalline anisotropy. A similar effect has been seen
in thin film SrRuO 3 [187], where it is suggested that the magnetocrystalline
anisotropy induces a crossover from Heisenberg to Ising behavior. The weak,
itinerant electron ferromagnet ZrZn 2 has mean field critical exponents β = 0.5
[93]. The most prominent theory on itinerant electron ferromagnetism by
4/3 4/3
Moriya [84, 85] predicts a TC - T dependence on the magnetization, which
is essentially β = 1.
The critical exponents γ and δ are also typical for large moment
ferromagnets (Fe, Ni and YIG [92] γ = 1.2 ± 0.2) as opposed to those found for
the weak, itinerant electron ferromagnet ZrZn 2 which has mean field critical
exponents γ = 1.0 and δ = 3 [93]. The three dimensional Ising and Heisenberg
models predict γ = 1.24, δ = 4.8 and γ = 1.39, δ = 4.8 respectively. The critical
exponents measured here also obey the scaling relation δ = 1 + γ/β.
163

164 Appendix A
1/χ 0 (emu -1 mol Gauss)
200
150
100
50
SrRuO 3 Polycrystal Inverse Susceptibility
Pellet
Single Crystal
0
150 200 250
Temperature (K)
300 350
Figure A- 7. Inverse magnetic susceptibility (1/χ = M/H) at H
= 10 kOe of polycrystalline SrRuO 3 compared to the single
crystal data from Figure A- 4. The solid line is the straightline
fit with T C = 165K which demonstrates the slightly
positive curvature of the data.
A positive curvature persists in the plot of 1/χ vs. T (Figure A- 7) even at
higher temperature where it should be linear for a Curie-Weiss ferromagnet.
Such a curvature can be caused by a temperature independent term in the
susceptibility χ = χ Const + C/(T - Θ), where C/(T - Θ) is the Curie-Weiss
susceptibility and is always positive (T > Θ). A χ Const < 0 of about -4 × 10 -4 emu
G -1 mol -1 will provide the observed curvature, and has been independently
observed elsewhere [175]. The temperature independent term χ Const , should
contain a positive contribution due to Pauli paramagnetism. This can be
estimated from measurements of the linear term of the specific heat [176],
giving χ Pauli ≈ +4 × 10 -4 emu G -1 mol -1 . The Landau diamagnetism should be

Critical Behavior and Anisotropy in Single Crystal SrRuO 3
negative and for simple band structures is smaller than χ Pauli (for free electrons
χ Landau = -χ Pauli /3). The core diamagnetism can be estimated from tables of
experimental values [76] giving χ Core = -0.7 × 10 -4 emu G -1 mol -1 . The sum of
these theoretical estimates χ Const = χ Pauli + χ Landau + χ Core is, however, still
positive while the experimental value appears to be negative. In the critical
region, 1/χ ∝ (T/T C - 1) γ with γ = 1.17, provides a positive curvature in the plot
of 1/χ vs. T. Since the mean field exponent γ = 1 is expected to be valid far
from T C , some other mechanism must provide the effective γ > 1 observed at
these higher temperatures.
Magnetic excitations which become thermally induced as the temperature
is raised above T = 0 reduce the magnetization from the ground state value.
The exponential decrease predicted by the mean field model has some
qualitative value but is never in good agreement with experiment. Collective
spin wave excitations and single particle (Stoner) excitations both decrease the
magnetization according to a power law M = M S (1 - (T/Θ n ) n ) which is in accord
with experiments where n ≈ 2 ± 1 and Θ n is of the order T C .
The limiting low T, H = 0 behavior of collective, spin wave excitations
(section 3.2.2.2.8) predicts n = 3/2 and for SrRuO 3 , Θ 3/2 ≈ 2.42T C = 400 K. The
Stoner theory of single (k-space) particle excitations (section 3.2.2.2.2) predicts
n = 2 and Θ 2 ≈ 1.41 T C (= 230 K for SrRuO 3 ). As the temperature and magnetic
field is raised, these models require further corrections. For example, higher
order corrections such as an additional n = 5/2 term in predicted in the spin
wave theory. For H ≠ 0, the low temperature spin-wave excitations become
quenched which can have the effect of increasing the average value on n [97].
The generalized model of spin fluctuations in ferromagnets ([84] section
165

166 Appendix A
T 3/2 Coefficient (10 -6 /K 3/2 )
200
150
100
50
0
A parameter in fit to M = M S (1 - AT 3/2 - BT 2 )
-50
0 25 50 75 100 125
Maximum Temperature of Fitting Range
150
3.2.2.2.3) includes both types of interacting magnetic excitations and predicts
n = 3/2 for H = 0, T = 0 but also n ≈ 2 in calculations [86].
T 2 Coefficient (10 -6 /K 2 )
25
20
15
Correlation of A and B
10
-20 0 20 40 60 80
T 3/2 Coefficient (10 -6 /K 3/2 )
Figure A- 8. Variation of the T 3/2 parameter in fitting the
magnetization data of single crystal SrRuO 3 to M = M S (1 -
AT 3/2 - BT 2 ) as the fitting range is increased. The upper inset
shows the correlation of the A and B parameters. In the
region where A is relatively stable (around T max = 60 K), A
decreases as T max is lowered. The symbols are the same as
those used in Figure A- 9.
Experimental results on metallic ferromagnets with substantial saturation
moments such as Fe, Ni [88, 189-192] and La 0.67 Sr 0.33 MnO 3 ([97], section 4.1.1)
can display n = 3/2 consistent with the spin wave stiffness determined by
neutron diffraction if significant corrections are included and only low
temperature data is analyzed. In contrast “weak” itinerant-electron

T 2 Coefficient (10 -6 /K 2 )
Critical Behavior and Anisotropy in Single Crystal SrRuO 3
50
40
30
20
10
-10
0 25 50 75 100 125 150
Maximum Temperature of Fitting Range
ferromagnets such as ZrZn 2 , Ni 3 Al and Sc 3 In, show n = 2 [86, 193-195] over a
wide temperature range.
0
As shown above (Figure A- 6) the magnetization of SrRuO 3 single crystal
can be well described by M = M S (1 - (T/Θ n ) n ) with n = 2 in the temperature
range of reliable measurement, 5 K to 100 K. A fit with n = 3/2 in this
temperature range is unsatisfactory.
B parameter in fit to M = M 0 (1 - AT 3/2 - BT 2 )
7T T 2 term
6T T 2 term
5T T 2 term
4T T 2 term
3T T 2 term
2T T 2 term
1T T 2 term
2kOe T 2 term
1kOe T 2 term
Figure A- 9. Variation of the T 2 parameter in fitting the
magnetization data of single crystal SrRuO 3 to M = M S (1 -
AT 3/2 - BT 2 ) as the fitting range is increased. In the region
where B is relatively stable (around T max = 60 K), B increases as
T max is lowered.
If one assumes n = 2 arises from Stoner excitations and collective
excitations result in n = 3/2, then one may expect the variation of
167

168 Appendix A
magnetization with temperature to fit with a combination of the n = 2 and
n = 3/2 terms, M = M S (1 - AT 3/2 - BT 2 ) [150] at low temperatures. Since an
additional parameter is included, an improvement of the fit does not
necessarily prove the significance of the added parameter. Furthermore, the
functions T 3/2 and T 2 are very similar making the fitting parameters highly
correlated.
The magnetization data for single crystal SrRuO 3 were fit to a polynomial
expression M = M S (1 - AT 3/2 - BT 2 ) from T = 2 K to T = T max . As expected, the A
and B parameters are highly correlated (Figure A- 8) with ∆B ≈ 2.2 × 10 -5 K -2 -
0.2 K -1/2 ∆A. The parameters A and B can then be plotted as a function of T max
(Figure A- 8 and Figure A - 9). For very low T max (< 30 K) the fit is unstable
since the difference in the magnetization at T

Critical Behavior and Anisotropy in Single Crystal SrRuO 3
T 2 Coefficient (10 -6 /K 2 )
50
40
30
20
10
0
B parameter in fit to M = M 0 (1 - BT 2 )
-10
0 25 50 75 100 125 150
Maximum Temperature of Fitting Range
In fitting to the data on SrRuO 3 single crystal the B parameter (Figure A-9)
is relatively stable and increases as lower temperature fitting ranges (T max ) are
used. The A parameter (Figure A- 8) is not as stable and appears to decrease as
T max is lowered. If this trend were to continue, then the T 3/2 contribution
would be very small at low temperatures. The removal of the AT 3/2 term
further stabilizes the B parameter (Figure A- 10), and makes B more
independent of T max than with AT 3/2 included. Thus, from this analysis of the
magnetization data, there is little evidence for a large AT 3/2 contribution to
the magnetization. This is in contrast with measurements on thin film [186,
187] which show T 3/2 dominating over T 2 .
Θ 2 (K)
240
235
230
233K = √2T
C
expected from
Θ (T = 60K)
2 max
225 simple Stoner Theory
220
Θ = B 2 1/2
215
210
0 1 2 3 4 5 6 7
Internal Field (T)
Figure A- 10. Variation of the T 2 parameter in fitting the
magnetization data of single crystal SrRuO 3 to M = M S (1 - BT 2 )
as the fitting range is increased. The parameter B for this fit is
more stable and constant than that shown in Figure A- 8.
Inset, variation of Θ 2 in a magnetic field.
The two corrections to the spin wave T 3/2 theory mentioned above (finite
magnetic field and higher order T (2n+1)/2 terms) may not adequately explain the
data presented here. The applied magnetic field will suppress spin wave
169

170 Appendix A
excitations for T < gµ B H/k B , resulting in an effective n > 3/2. If this were a
significant effect, the T 3/2 fitting parameter A should decrease as H is increased
particularly as T max approaches 0 K. Such a systematic decrease in A is not
obviously apparent (Figure A- 8). One cannot however, exclude the
demagnetization or anisotropy field which are present even when the applied
H = 0.
AT 3/2 Coefficient (10 -6 /K 3/2 )
140
120
100
80
60
40
20
M = 1 - (T/400 K) 3/2 - (T/600 K) 5/2
fit with M = M (1 - AT
S 3/2 - BT2 )
A
B
0
0.0
0 20 40 60 80 100
Maximum Temperature (K) of Fitting Range
Figure A- 11. Variation of A and B fitting parameters in the
hypothetical case where the true magnetization is given by T 3/2
and T 5/2 terms.
If the true magnetization were a sum of T 3/2 and T 5/2 terms, M/M S =
(1 - (T/400 K) 3/2 - (T/600 K) 5/2 ) for instance, the data would fit well to M/M S =
(1 - AT 3/2 - BT 2 ). The fitting parameters A and B as a function of T max for this
example are shown in Figure A- 11. The A parameter is relatively constant,
increasing slightly as T max approaches 0 K. In the limit T max approaches 0 K,
A = 125 × 10 -6 K -3/2 or Θ 3/2 = 400.3 K, which is very close to the true value of
400 K. The B fitting parameter is not nearly as stable and decreases as T max
2.0
1.5
1.0
0.5
BT 2 Coefficient (10 -6 /K 2 )

Critical Behavior and Anisotropy in Single Crystal SrRuO 3
approaches 0 K. Since the exponent of the BT 2 fit (2) is smaller than 5/2,
which is the exponent used in the example, a larger B is needed to fit the data
at higher temperatures. Thus, a dramatically decreasing B as T max approaches
0 K, is indicative of a fitting exponent (2 in this example) smaller than the
true exponent (5/2 in this example).
The data on SrRuO 3 shows the opposite effect: B is relatively constant
while A decreases as T max approaches 0 K. This decrease of A indicates that
the exponent 3/2 is smaller than the true exponent, and therefore, the
addition of higher order terms to the spin wave form of the magnetization
will not fully explain the data on SrRuO 3 .
Positive B (and A) parameters can be converted to the Θ n values used
above via Θ 2 = B -1/2 and Θ 3/2 = A -2/3 . The Θ 2 values extracted from fits
described above are shown as a function of the internal magnetic field in
Figure A- 10. The values of Θ 2 (H) extrapolated to T max = 0 are 20 K less than
the corresponding Θ 2 (H) for T max = 60 K. The single crystal Θ 2 parameters
increase only slightly as the field is increased, about 2.6 K/Tesla. These values
for Θ 2 are consistent with measurements of the field dependent heat capacity
of polycrystalline SrRuO 3 (Figure B- 3) via the Maxwell relation
⎛ ∂CH ⎞
⎝ ∂H ⎠ T
= T ∂ 2 M
∂T 2
⎛ ⎞
⎜ ⎟ .
⎝ ⎠ H
Conclusion
Various magnetic properties of single crystal SrRuO 3 along the magnetic
easy direction have been measured. The saturation moment, 1.61µ B /Ru, is
larger than that reported previously for single crystal material but in accord
with experiments on polycrystalline material and theoretical calculations.
The crystals show extremely large cubic magnetocrystalline anisotropy while
no substantial uniaxial magnetocrystalline anisotropy. The critical exponent
171

172 Appendix A
β changes from Heisenberg like 0.39 near T C to Ising like 0.32 further from T C .
The susceptibility exponent γ = 1.17 persists to 2 T C , where it is expected to
decrease to 1. The magnetization from 5 K to 100 K is well described by
M = M S (1 - (T/Θ n ) n ) where n ≈ 2. The addition of a T 3/2 term is not obviously
significant.

Appendix B. Magnetic Excitations and Specific Heat in
SrRuO 3
Low temperature magnetic excitations can be probed not only by
measurements of the magnetization but also by the heat capacity. As will be
discussed below, the two measurements are related via a Maxwell relation.
SrRuO 3 polycrystalline samples were prepared by repeated heating of a R u
and SrCO 3 mixture at 1260°C (seciton 2.1.1). Heat capacity data were taken
using the relaxation method [102] (section 3.3). Constant field data were taken
on a zero field cooled sample at 0 and 8 T, and in zero field after the
application of an 8 T field. In addition, data at temperatures between 4 K and
5 K were taken at several intermediate fields.
Specific heat data taken in 0 T and 8 T applied magnetic field are shown in
Figure B- 1. The zero field cooled and zero field remnant data are equal
within experimental error. Between 3 K and 13 K both the 0 T and 8 T data
appear to be well described by a straight line on this plot of c/T vs. T 2 ,
suggesting that the specific heat can be described by γT+βT 3 . The parameter
values from fitting the data in zero field are γ = 37 mJ mol -1 K -2 and
β = 0.19 mJ mol -1 K -4 . This value of β corresponds to a Debye temperature of
371 K and is in good agreement with previous zero-field measurements [176,
196], while γ is about 20% larger than that found previously. The value of γ in
zero field, 37 mJ mol -1 K -2 , is 4.6 times that expected of the bare density of states
given by spin resolved band structure calculations [176]. The effect of
demagnetization fields should be negligible since the correction to the specific
heat is approximately proportional to T 3 but only about 0.1% of the measured
value of β [197, 198].
173

174 Appendix B
c P,H /T (mJ/mol·K 2 )
110
100
90
80
70
60
50
40
SrRuO 3 Specific Heat
0 T zfc
0 T rem
8 T
-5
0 50 100 150 200 250 300 350
Temperature 2 (K 2 )
30
0 50 100 150 200 250 300 350
Temperature 2 (K 2 )
Above 13 K and below 3 K the data in Figure B- 1 deviate from a straight
line. The deviation above 13K can be fit by an expression for the specific heat
of a thermally activated Einstein (optical) phonon mode, with an Einstein
temperature of Θ Ε = 140 K. Both β and Θ Ε are field independent within
experimental error. The temperature range below 3 K is not large enough to
draw conclusion about the functional form of the low-temperature upturn.
A field independent term proportional to T 3 ln(T) is expected from the non-
spinwave spin fluctuations [91, 194, 199]. A Shottky anomaly due to free
∆c P,H /T (mJ/mol·K 2 )
1
0
-1
-2
-3
-4
c P,H /T (rem) - c P,H /T (zfc)
c P,H /T (8T) - c P,H /T (zfc)
Figure B- 1 Heat capacity of SrRuO 3 cooled in zero field (zfc),
in an 8 T magnetic field, and in zero field after being
magnetized (rem). Inset, difference between the heat capacity
measured after cooling in zero field with that in 8 T and the
remnant magnetized state.

c P,H /T (mJ/mol·K)
65
60
55
50
45
40
Specific Heat of SrRuO 3
SrRuO 3 Specific Heat in Zero Field
c P,H (T) = γT + βT 3 +
2mJ/(mol·K 5/2 ) T 3/2
c P,H (T) = γT + βT 3
H=0 Remnant
H=0 Zero Field Cooled
35
0 25 50 75 100 125 150
Temperature 2 (K 2 )
Figure B- 2. Zero field heat capacity data fit with two free
parameters, γ and β. Solid line, including the T 3/2 contribution
expected theoretically for spin waves (see text). Dashed line,
without any T 3/2 contribution.
paramagnetic spins will give a low temperature upturn but should also be
noticeable as Curie paramagnetism in magnetization measurements.
Adding a positive T 3/2 term to the model for the specific heat, as expected
for pure spin wave excitations (3.2.2.2.8) [95], does not improve the fit (Figure
B- 2). Assuming 6 nearest neighbors, spin 1, and T C = 163 to determine J, this
value is A = 2mJ mol -1 K -5/2 . Previous magnetization measurements on thin
films suggest that this term is correct within a factor of 2 [186]. Using a
functional form of γT + AT 3/2 + βT 3 , the coefficient A is less than 1.2 (0 T zfc),
2.5 (8 T), and 0.6 (0 T rem) mJ mol -1 K -5/2 for each of the three data sets i n
175

176 Appendix B
Figure 3 at the 90% statistical confidence level. The T 3/2 term in Fe and Ni is
quite small but has nevertheless been extracted from the heat capacity data
[200-202].
γ(H) (mJ/mol-K 2 )
Field Dependence of the Density of States in SrRuO 3
38
37
36
35
34
34
33
32
Predicted from M vs. T
Single Crystal Data
0 2 4 6 8
Magnetic Field (Tesla)
Figure B- 3. The linear term of the heat capacity γ as a
function of magnetic field. Each circle is from a single c P,H
datum between 4.3 and 5 K with phonons subtracted: γ(H) =
(c P,H (T) - βT 3 )/T with β = 0.191 mJ/mol·K 4 . γ(H) for each square
was determined by fitting 15-20 data points between 6 and 12 K
to: c P,H (T) = γT + βT 3 . The triangles are calculated from the
magnetization data of a single crystal.
The linear-T term, γ, is significantly depressed in an 8 T magnetic field.
The inset to Figure B- 1 shows the difference between the zero-field cooled
data in zero field and each of the other two data sets, plotted as ∆c/T. The data

Specific Heat of SrRuO 3
taken in 0 T after application of an 8 T field (circles) shows that the zero-field
data are not hysteretic. On this plot of ∆c/T, the 8 T data (diamonds) show
that the only observed change in the specific heat is a decrease in the linear-T
term.
To further understand the field dependence of γ, specific heat points
between 4 K and 5 K at several intermediate fields were also taken. These
data are shown in Figure B- 3 (open circles) as γ(H) = (c(T,H) - βT 3 )/T using the
value of β measured at 0 T and 8 T. On the same plot, the open squares
indicate the fitted value of γ from the three data sets in Figure B- 1.
Figure B- 3 shows that the coefficient of the linear-T term in the specific heat
decreases approximately linearly with applied magnetic field, H.
To summarize, the specific heat between 3 and 13 K can be described by
c(T,H) = (γ - γ ´H)T+βT 3 .
Although the magnetization of polycrystalline SrRuO 3 is irreversible, the
measured heat capacity is not. Therefore, the heat capacity originates from
processes reversible with respect to a magnetic field, showing that the entropy
associated with random domains is negligible. In this sense, the heat capacity
is a better thermodynamic measure of the low-lying excitations. The
magnetic field dependence of the heat capacity and the magnetization are
related by the following Maxwell relation:
⎛ ∂CH ⎞
⎝ ∂H ⎠ T
= T ∂ 2 M
∂T 2
⎛ ⎞
⎜ ⎟
⎝ ⎠ H
(1)
According to equation 1, the first derivative of the heat capacity with
respect to field gives information about the second derivative of the
magnetization with respect to temperature. This relation is generally valid
for an isotropic system as long as there are no irreversible processes involved
and no PV work is done. Equation 1 is actually three equations since H and M
177

178 Appendix B
are vectors. Thus for an ideal, simple experiment, the sample would be very
soft magnetically (no magnetic hysteresis) and the magnetic field would be
aligned in the magnetic easy direction to keep M and H parallel. If possible,
the magnetic and heat capacity experiments should be done on the same
sample in the same direction; although, in principle different samples should
give the same result since both M and C P in a reversible sample should not
depend on the microstructure. In order to use equation 1 to relate the
reversible magnetization from the measured specific heat, it is assumed that
effects due to the field dependence of the magnetocrystalline anisotropy are
insignificant.
The simplest spin wave model (section 3.2.2.2.8) predicts a T 3/2 dependence
of the magnetization at low temperatures:
MHT ( , ) T
≈1− MH ( , 0)
3 ( H)
⎛ ⎞
⎜ ⎟
⎝ Θ ⎠
with Θ 3/2 ≈ 2.42T C = 400K [95, 186] for SrRuO 3 . Corrections due to H ≠ 0 and
T > 0 can also be calculated [95]. The Stoner/Wohlfarth theory of magnetism
(section 3.2.2.2.2) [81] predicts a somewhat different temperature dependence:
MHT ( , ) T
≈1− MH ( , 0)
( H)
⎛ ⎞
⎜ ⎟
⎝ Θ ⎠
(Θ 2 ≈ √2T C = 231K for SrRuO 3 ) due to single (k-space) particle excitations.
The Maxwell relation together with the observed magnetic field
dependence of the specific heat above 4 K, γ ´HT, implies that the second
2
⎛ ∂ M⎞
derivative of the magnetization with respect to temperature, ⎜ 2 ⎟ , is a
⎝ ∂T
⎠ H
constant. This result is consistent with the dominant T 2 temperature
dependence of the reversible discussed above.
In order to make quantitative comparisons between the specific heat data
and the magnetization data, equation 1 can be integrated with respect to H
2
2
3 2
2
(2)
(3)

Specific Heat of SrRuO 3
from H = 0 to H = ∆H. Since magnetization data is not available at all fields,
the integral is approximated to a finite sum with step size dH:
∆H/
dH−
∆CH(
T)
dH M( n ⋅ dH, T)
≈ ∑ T∆H ∆Hn=
T
∂
1 2
(4)
2
0 ∂
where ∆CH (T)=CH (T,H=∆H)-CH (T,H=0). The two simplest physically plausible
temperature dependencies of the reversible magnetization, T 2 (equation 3) or
T 3/2 (equation 2), each correspond to a simple temperature dependence of
∆CH
T
T∆H ( ) :
H/ dH 1
MHT ( , ) T CH( T)
2dH Mn ( dH,
0)
≈1− Const.
0
2
MH ( , 0)
( H)
T H H ( n dH)
⎛
∆ −
⎞ ∆ − ⋅
⎜ ⎟ ⇒ = ∑
= <
⎝ Θ ⎠ ∆ ∆ Θ ⋅
2
2
H/ dH 1
MHT ( , ) T CH( T)
3dH
Mn ( dH,
0) 1
≈1− 3
2
MH ( , 0)
3 ( H)
T H 4 T H n 0 3 ( n dH) T
⎛ ⎞
− ⋅
⎜ ⎟ ⇒ = ∝
⎝ ⎠
⋅
−
∆ −
∆
∑
Θ
∆ ∆ = Θ
2
3 2
Since the ∆cPH
, T
T∆H ( ) data of Figure B- 1 are constant in temperature, these
data can be fit to a T 2 dependence of the magnetization, but not by T 3/2 alone.
Using the parameters from the fits to the magnetization data of single crystal
SrRuO3 (Appendix A) gives quantitative values for the constant ∆cPH
, T
T∆H ( ) =
-0.42 mJ mol -1 K -2 T -1 (about 20% less than that measured by ∆cPH
, T
T∆H ( ) ). A field
independent Θ2 (H) is consistent with γ decreasing linearly with magnetic
field. Since the two measurements are on different types of samples with
different orientations; nevertheless this result illustrates the potential for this
method.
The failure to observe a T 3/2 dependence does not necessarily eliminate the
possibility of spin wave excitations in SrRuO 3 ; the data could be easily
described within the spin wave theory (section 3.2.2.2.8) if additional
n=
0
2
2
179
(5)
(6)

180 Appendix B
corrections are included. Using these corrections, the predicted
magnetization fit to T n will necessarily have n ≈ 2 to fit the data.
At low temperatures and finite fields, a gap is expected in the spin wave
dispersion [95], predicting an exponential temperature dependence of the
magnetization at the lowest temperatures. Combined with the spin wave n =
3/2, H = 0 form, this effectively increases n [97]. The inclusion of this gap
affects the predicted ∆cPH
, T
T∆H ( ) primarily at low temperatures, as is shown in
(Figure 3-11).
The linear term in the specific heat γ is usually interpreted in terms of the
(nonmagnetic) electronic contribution to the specific heat. Spin fluctuations
are predicted to enhance γ in ferromagnetic and nearly ferromagnetic metals
[84]. For γ = γ ele (1 + λ e-p + λ mag ) where γ ele is γ expected from band calculations
and λ e-p ≈ 0.5 is the electron-phonon mass enhancement [176]. From these
values λ mag ≈ 3.1, the electron-spin fluctuation mass enhancement, is
estimated. Theoretically, in the presence of a magnetic field, spin fluctuations
are suppressed and therefore γ will decrease [203, 204]. From equation 1, these
spin fluctuations which cause the field dependence of γ are required to give a
T 2 dependence of the magnetization, thereby unifying the two interpretations
for the field dependence of γ. A decrease in the coefficient of the linear-T
term, γ, has also been observed in the weak itinerant ferromagnets ZrZn 2 and
Sc 3 In [199, 205] (which also show a T 2 dependence of the magnetization).
In conclusion, the magnetic field dependence of the heat capacity provides
a useful method for studying the low-lying excitations, and for determining
the temperature dependence of the magnetization via thermodynamic
relations. For a ferromagnetic substance the magnetic field dependence of the
low-temperature heat capacity can be a more sensitive probe of the low-

Specific Heat of SrRuO 3
energy magnetic excitations than the saturation magnetization, particularly
for polycrystalline samples with large magnetocrystalline anisotropy such as
SrRuO 3 .
Like the magnetization measurements, the heat capacity measurements
can be well described by a decrease in the magnetization proportional to T n
where n ≈ 2. This can be understood by any of three mechanisms. First, if the
corrections are included in the spin wave theory then an effective n > 3/2 is
predicted. Second, single particle excitations are present. Third, the
excitations interact (requiring a more generalized description [85]) to give an
effective exponent n ≈ 2.
The same phenomena can also be interpreted as a spin fluctuation
enhancement of the electronic specific heat, which can be suppressed by a
magnetic field.
The help of K. A. Moler for the heat capacity measurements is greatly
appreciated.
181

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